NCERT Exemplar Class 12 Maths Solutions Chapter 5 Continuity and Differentiability

Q: What is the difference between continuity and differentiability?

Continuity and differentiability are closely related but distinct concepts in calculus. A function is continuous at a point if there is no break, jump, or hole, meaning the graph flows smoothly through that point. On the other hand, a function is differentiable at a point if it has a defined derivative there, implying the graph has a well-defined tangent and no sharp corners or cusps. All differentiable functions are continuous, but not all continuous functions are differentiable. For example, the absolute value function is continuous everywhere but not differentiable at x=0 due to a sharp corner.

Q: Can a function be differentiable but not continuous?

No, a function cannot be differentiable without being continuous. Differentiability implies continuity. If a function is differentiable at a point, it must also be continuous at that point. This is because finding a derivative involves calculating a limit, and for that limit to exist, the function must not break or jump at that point. However, the reverse is not always true, a function can be continuous but not differentiable, such as the absolute value function at x=0, which is continuous but has a sharp corner, making it non-differentiable at that poin

Q: What is the significance of Rolle’s theorem in differentiability?

Rolle's Theorem is significant in understanding the behavior of differentiable functions. It states that if a function is continuous on [a, b], differentiable on (a, b), and f(a)=f(b), then there exists at least one point c in (a, b) where f'(c)=0. This means the function has a horizontal tangent (slope zero) at some point between a and b. Rolle's Theorem helps in proving other important results like the Mean Value Theorem and is useful in analyzing turning points, ensuring the existence of critical points in various real-life applications involving motion or optimization.

Q: What is the concept of logarithmic differentiation?

Logarithmic differentiation is a technique used to differentiate complex functions, especially those involving products, quotients, or variables raised to variable powers. The process involves taking the natural logarithm (ln) of both sides of the function, simplifying using logarithmic identities, and then differentiating implicitly. This method is particularly useful when dealing with functions like y=x x or y=(x 2 +1) x , where standard rules are difficult to apply directly. Logarithmic differentiation simplifies calculations, making it easier to handle functions with multiple terms or exponents, and is a powerful tool in both calculus and applied mathematics.

Q: Why is every differentiable function always continuous, but not vice versa?

Every differentiable function is always continuous because differentiability requires the function's limit to exist and match the function's value at a point. In other words, for a function to be differentiable, it must first be smooth and unbroken-i.e., continuous. However, not all continuous functions are differentiable. A function can be continuous without having a defined derivative at some points, especially where there are sharp turns. For example, the absolute value function |x| is continuous everywhere but not differentiable at x=0, since it has a sharp corner at that point.

Team Careers360Updated on 02 Apr 2025, 11:47 PM IST

Limits and differentiability are fundamental concepts in calculus that help us understand change and continuity in mathematical functions. You can have a question about what the limit tells us. A limit describes the value that a function approaches as the input (variable) approaches a certain point, while differentiability indicates whether a function is differentiable at that point. These concepts are essential for analyzing motion, growth, and change in various fields. For example, in real life, the speed of a moving car at a specific instant is found using differentiation by calculating the limit of the average speed as the time interval approaches zero. The NCERT Class 12 Mathematics book covers everything that will come in the board exams and helps in creating a base of the topics.

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Class 12 Maths Chapter 5 exemplar solutions Exercise: 5.3
Page number: 107-116
Total questions: 106

Question:1

Examine the continuity of the function
$f(x) = x^3 + 2x - 1 $at x = 1$

Answer:

We have, $f(x)=x^3+2 x^2-1$

For continuity at $\mathrm{x}=1$

$\begin{aligned} & \therefore \text { R.H.L. }=\lim _{x \rightarrow 1^{+}} f(x) \\ & =\lim _{h \rightarrow 0} f(1+\mathrm{h}) \\ & =\lim _{\mathrm{h} \rightarrow 0}\left[(1+\mathrm{h})^3+2(1+\mathrm{h})^2-1\right]=2\end{aligned}$

And L.H.L. $=\lim _{x \rightarrow 1^{-}} \mathrm{f}(x)$

$\begin{aligned} & =\lim _{h \rightarrow 0} f(1-h) \\ & =\lim _{h \rightarrow 0}\left[(1-h)^3+2(1-h)^2-1\right]=2\end{aligned}$

Also $f(1)=1+2-1=2$

Thus $\lim _{x \rightarrow 1^{+}} \mathrm{f}(x)=\lim _{x \rightarrow 1^{-}} \mathrm{f}(x)=\mathrm{f}(1)$

Thus $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=1$

Question:2

Find which of the functions is continuous or discontinuous at the indicated points:
$f(x)=\left\{\begin{array}{cl} 3 x+5, & \text { if } x \geq 2 \\ x^{2}, & \text { if } x<2 \end{array}\right. \text { at } x=2$

Answer:

Given,
$f(x)=\left\{\begin{array}{cl} 3 x+5, & \text { if } x \geq 2 \\ x^{2}, & \text { if } x<2 \end{array}\right.$
We need to check its continuity at x = 2
A function f(x) is said to be continuous at x = c if,
Left-hand limit (LHL at x = c) = Right-hand limit (RHL at x = c) = f(c).
Mathematically, we can represent it as
$\lim _{h \rightarrow 0} f(c-h)=\lim _{h \rightarrow 0} f(c+h)=f(c)$
Where h is a very small number very close to 0 (h→0)
Now, according to the above theory-
f(x) is continuous at x = 2 if -
$\begin{aligned} &\lim _{h \rightarrow 0} \mathrm{f}(2-\mathrm{h})=\lim _{h \rightarrow 0} \mathrm{f}(2+\mathrm{h})=\mathrm{f}(2)\\ & \text { Now we can see that, }\\ &LH L=\lim _{h \rightarrow 0} f(2-h)=\lim _{h \rightarrow 0}(2-h)^{2} \text {Using equation } \\ & \therefore L H L=(2-0)^{2}=4 \ldots (2) \end{aligned}$
Similarly, we proceed with RHL-
$\begin{aligned} &\lim _{\mathrm{RHL}}=\underset{\mathrm{h} \rightarrow 0}{\mathrm{f}(2+\mathrm{h})}=\lim _{\mathrm{h} \rightarrow 0}\{3(2+\mathrm{h})+5\}\\ &\therefore \mathrm{RHL}=3(2+0)+5=11 \ldots(3)\\ &\text { then, }\\ &f(2)=3(2)+5=11 \ldots(4) \end{aligned}$
Now, from equations 2, 3, and 4, we can conclude that
$\begin{aligned} &\lim _{h \rightarrow 0} f(2-h) \neq \lim _{h \rightarrow 0} f(2+h)\\ &\therefore f(x) \text { is discontinuous at } x=2 \end{aligned}$

Question:3

Find which of the functions is continuous or discontinuous at the indicated points:
$f(x)=\left\{\begin{array}{cl} \frac{1-\cos 2 x}{x^{2}}, & \text { if } x \neq 0 \\ 5, & \text { if } x=0 \end{array}\right. \text { at } x=0$

Answer:

Given,
$f(x)=\left\{\begin{array}{cl} \frac{1-\cos 2 x}{x^{2}}, & \text { if } x \neq 0 \\ 5, & \text { if } x=0 \end{array}\right.$
We need to check its continuity at x = 0
A function f(x) is said to be continuous at x = c if,
Left-hand limit (LHL at x = c) = Right-hand limit (RHL at x = c) = f(c).
Mathematically, we can present it as
$\lim _{h \rightarrow 0} \mathrm{f}(\mathrm{c}-\mathrm{h})=\lim _{\mathrm{h} \rightarrow 0} \mathrm{f}(\mathrm{c}+\mathrm{h})=\mathrm{f}(\mathrm{c}) \\$

Where $h$ is a very small number very close to $0(\mathrm{~h} \rightarrow 0)$ Now according to above theory-$f(x)$ is continuous at $x=0$ if

$ \lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0} f(0+h)=f(0) $
$\begin{aligned} &\text { then, }\\ &\lim _{\mathrm{h} \mathrm{HL}=\mathrm{h} \rightarrow 0} \mathrm{f}(-\mathrm{h})=\lim _{\mathrm{h} \rightarrow 0} \frac{1-\cos 2(-\mathrm{h})}{(-\mathrm{h})^{2}} \underbrace{\{\text { using equation } 1\}}\\ &\text { As we know } \cos (-\theta)=\cos \theta\\ &\Rightarrow \mathrm{LHL}=\lim _{\mathrm{h} \rightarrow 0} \frac{1-\cos 2 \mathrm{~h}}{\mathrm{~h}^{2}}\\ &\because 1-\cos 2 x=2 \sin ^{2} x\\ &\therefore \mathrm{lim}_{\mathrm{h} \rightarrow 0} \frac{2 \sin ^{2} \mathrm{~h}}{\mathrm{~h}^{2}}=2 \lim _{\mathrm{h} \rightarrow 0}\left(\frac{\sin \mathrm{h}}{\mathrm{h}}\right)^{2} \end{aligned}$
This limit can be evaluated directly by putting the value of h because it is taking the indeterminate form (0/0)
As we know,
$\begin{aligned} &\lim _{x \rightarrow 0} \frac{\sin x}{x}=1\\ &\therefore \mathrm{LHL}=2 \times 1^{2}=2 \ldots(2)\\ &\text { Similarly, we proceed for RHL- }\\ &\lim _{\mathrm{RHL}} \mathrm{h} \rightarrow 0^+{\mathrm{f}}(\mathrm{h})=\lim _{\mathrm{h} \rightarrow 0} \frac{1-\cos 2(\mathrm{~h})}{(\mathrm{h})^{2}}\\ &\Rightarrow \mathrm{RHL}=\lim _{\mathrm{h} \rightarrow 0} \frac{1-\cos 2 \mathrm{~h}}{\mathrm{~h}^{2}}\\ &\Rightarrow \mathrm{RHL}=\lim _{\mathrm{h} \rightarrow 0} \frac{2 \sin ^{2} \mathrm{~h}}{\mathrm{~h}^{2}}=2 \lim _{\mathrm{h} \rightarrow 0}\left(\frac{\sin \mathrm{h}}{\mathrm{h}}\right)^{2} \end{aligned}$
Again, using the sandwich theorem, we get -
$RHL = 2 \times 1^2 = 2...(3)$
And,
f (0) = 5 …(4)
From equations 2, 3, and 4 we can conclude that
$\lim _{h \rightarrow 0} \mathrm{f}(0-\mathrm{h})=\lim _{\mathrm{h} \rightarrow 0} \mathrm{f}(0+\mathrm{h}) \neq \mathrm{f}(0)$
∴ f(x) is discontinuous at x = 0

Question:4

Find which of the functions is continuous or discontinuous at the indicated points:
$f(x)=\left\{\begin{aligned} \frac{2 x^{2}-3 x-2}{x-2}, \text { if } & x \neq 2 \\ 5,\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: & \text { if } x=2 \end{aligned}\right.$ at x = 2.

Answer:

Given,
$f(x)=\left\{\begin{aligned} \frac{2 x^{2}-3 x-2}{x-2}, \text { if } & x \neq 2 \\ 5,\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: & \text { if } x=2 \end{aligned}\right.$
We need to check its continuity at x = 2
A function f(x) is said to be continuous at x = c if,
Left-hand limit(LHL at x = c) = Right-hand limit(RHL at x = c) = f(c).
Mathematically, we can represent it as
$\lim _{h \rightarrow 0} f(c-h)=\lim _{h \rightarrow 0} f(c+h)=f(c)$
Where h is a very small number very close to 0 (h→0)
Now, according to the above theory-
f(x) is continuous at x = 2 if -
$\lim _{h \rightarrow 0} f(2-h)=\lim _{h \rightarrow 0} f(2+h)=f(2)$
Then,
$\begin{aligned} &\lim _{h \rightarrow 0} f(2-h)=\lim _{h \rightarrow 0} \frac{2(2-h)^{2}-3(2-h)-2}{(2-h)-2} \quad\{\text { using equation } 1\}\\ &\Rightarrow \mathrm{LHL}=\lim _{\mathrm{h} \rightarrow 0} \frac{2\left(4+\mathrm{h}^{2}-4 \mathrm{~h}\right)-6+3 \mathrm{~h}-2}{-\mathrm{h}}\\ &\Rightarrow \mathrm{lim} \frac{8+2 \mathrm{~h}^{2}-8 \mathrm{~h}-6+3 \mathrm{~h}-2}{-\mathrm{h}}=\lim _{\mathrm{h} \rightarrow 0} \frac{2 \mathrm{~h}^{2}-5 \mathrm{~h}}{-\mathrm{h}}\\ &\Rightarrow \mathrm{LHL}=\lim _{\mathrm{h} \rightarrow 0} \frac{-\mathrm{h}(5-2 \mathrm{~h})}{-\mathrm{h}}=\lim _{\mathrm{h} \rightarrow 0}(5-2 \mathrm{~h})\\ &\therefore \mathrm{LHL}=5-2(0)=5 \ldots(2) \end{aligned}$
Similarly, we proceed with RHL-

$ \lim _{\mathrm{RHL}=\mathrm{h} \rightarrow 0} \mathrm{f}(2+\mathrm{h})=\lim _{\mathrm{h} \rightarrow 0} \frac{2(2+\mathrm{h})^{2}-3(2+\mathrm{h})-2}{(2+\mathrm{h})-2} \text { [using equation } \left.1\right\} $
$\Rightarrow \mathrm{RHL}=\lim _{\mathrm{h} \rightarrow 0} \frac{2\left(4+\mathrm{h}^{2}+4 \mathrm{~h}\right)-6-3 \mathrm{~h}-2}{\mathrm{~h}}$$\Rightarrow \mathrm{RHL}=\lim _{\mathrm{h} \rightarrow 0} \frac{8+2+8 \mathrm{~h}-6-3 \mathrm{~h}-2}{-\mathrm{h}}=\lim _{\mathrm{h} \rightarrow 0} \frac{2 \mathrm{~h}^{2}+5 \mathrm{~h}}{\mathrm{~h}}$

$\Rightarrow \lim _{h \rightarrow 0} \frac{h(5+2 h)}{h}=\lim _{h \rightarrow 0}(5+2 h)$

$\therefore \mathrm{RHL}=5+2(0)=5 \ldots(3)$

And, $f(2)=5\{$ using equation 1$\} \ldots(4)$
From the above equations 2, 3, and 4, we can say that

$\lim _{h \rightarrow 0} f(2-h)=\lim _{h \rightarrow 0} f(2+h)=f(2)=5$
∴ f(x) is continuous at x = 2

Question:5

Find which of the functions is continuous or discontinuous at the indicated points:
$f(x)=\left\{\begin{array}{cl} \frac{|x-4|}{2(x-4)}, \text { if } & x \neq 4 \\ 0, & \text { if } x=4 \end{array}\right.$
at x = 4

Answer:

Given,
$f(x)=\left\{\begin{array}{cl} \frac{|x-4|}{2(x-4)}, \text { if } & x \neq 4 \\ 0, & \text { if } x=4 \end{array}\right.$ ...(1)
We need to check its continuity at x = 4
A function f(x) is said to be continuous at x = c if,
Left-hand limit(LHL at x = c) = Right-hand limit(RHL at x = c) = f(c).
Mathematically, we can represent it as
$\lim _{h \rightarrow 0} f(c-h)=\lim _{h \rightarrow 0} f(c+h)=f(c)$
Where h is a very small number very close to 0 (h→0)
Now, according to the above theory-
f(x) is continuous at x = 4 if -
$\lim _{h \rightarrow 0} f(4-h)=\lim _{h \rightarrow 0} f(4+h)=f(4)$
Clearly,

$\begin{aligned} &\lim _{\mathrm{h} \mathrm{HL}} \mathrm{f}(4-\mathrm{h})=\lim _{\mathrm{h} \rightarrow 0} \frac{|4-\mathrm{h}-4|}{2(4-\mathrm{h}-4)}\{\text { using equation } 1\}\\ &\Rightarrow \mathrm{LHL}=\lim _{\mathrm{h} \rightarrow 0} \frac{|-\mathrm{h}|}{-2 \mathrm{~h}}\\ &\because \mathrm{h}>0 \text { as defined above. }\\ &\therefore|-h|=h\\ &\Rightarrow \mathrm{LHL}=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{h}}{-2 \mathrm{~h}}=-\frac{1}{2} \lim _{\mathrm{h} \rightarrow 0} 1\\ &\therefore \mathrm{LHL}=-1 / 2 \ldots(2) \end{aligned}$
Similarly, we proceed with RHL-
$\begin{aligned} &{\mathrm{RHL}}=\lim _{\mathrm{h}\rightarrow 0}{\mathrm{f}}(4+\mathrm{h})=\lim _{\mathrm{h} \rightarrow 0} \frac{|4+\mathrm{h}-4|}{2(4+\mathrm{h}-4)}\{\text { using equation } 1\}\\ &\Rightarrow \mathrm{RHL}=\lim _{\mathrm{h} \rightarrow 0} \frac{|\mathrm{h}|}{2(\mathrm{~h})}\\ &\because \mathrm{h}>0 \text { as defined above. }\\ &\therefore|h|=h \end{aligned}$
$\Rightarrow \mathrm{RHL}=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{h}}{2 \mathrm{~h}}=\frac{1}{2} \lim _{\mathrm{h} \rightarrow 0} 1$
$\therefore \mathrm{RHL}=1 / 2 \ldots(3)$ And, $f(4)=0\{$ using eqn 1$\} \ldots(4)$

From equations 2, 3, and 4, we can conclude that

$ \lim _{h \rightarrow 0} f(4-h) \neq \lim _{h \rightarrow 0} f(4+h) \neq f(4) $

∴ f(x) is discontinuous at x = 4

Question: 6

$f(x)=\left\{\begin{array}{cl} |x| \cos \frac{1}{x}, \text { if } & x \neq 0 \\ 0, & \text { if } x=0 \end{array}\right.$
at x = 0

Answer:

We have, $\begin{cases}|x| \cos \frac{1}{x}, & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{cases}$

At $\mathrm{x}=0$

$\begin{aligned} & \text { L.H.L. }=\lim _{x \rightarrow 0^{-}}|x| \cos \frac{1}{x} \\ & =\lim _{\mathrm{h} \rightarrow 0}|0-\mathrm{h}| \cos \frac{1}{0-\mathrm{h}} \\ & =\lim _{\mathrm{h} \rightarrow 0} \mathrm{~h} \cos \frac{1}{\mathrm{~h}} \\ & =0 \times[\text { an oscillating number between }-1 \text { and } 1]=0\end{aligned}$
$\begin{aligned} & \text { R.H.L. }=\lim _{x \rightarrow 0^{+}}|x| \cos \frac{1}{x} \\ & =\lim _{\mathrm{h} \rightarrow 0}|0+\mathrm{h}| \cos \frac{1}{0+\mathrm{h}} \\ & =\lim _{\mathrm{h} \rightarrow 0} \mathrm{~h} \cos \frac{1}{\mathrm{~h}} \\ & =0 \times[\text { an oscillating number between }-1 \text { and } 1]=0\end{aligned}$
Also $\mathrm{f}(0)=0 \quad \ldots$. (Given)

Thus, L.H.L. $=$ R.H.L. $=f(0)$

So, $f(x)$ is continuous at $x=0$

Question:7

Find which of the functions is continuous or discontinuous at the indicated points:
Check continuity at x =a $f(x)=\left\{\begin{array}{cl} |x-a| \sin \frac{1}{x-a}, \text { if } & x \neq a \\ 0, & \text { if } x=a \end{array}\right.$

Answer:

We have, $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{ll}|x-\mathrm{a}| \sin \frac{1}{x-\mathrm{a}}, & \text { if } x \neq 0 \\ 0, & \text { if } x=\mathrm{a}\end{array}\right.$ at $\mathrm{x}=\mathrm{a}$ At $\mathrm{x}=\mathrm{a}$

$\begin{aligned} & \text { L.H.L. }=\lim _{x \rightarrow \mathrm{a}^{-}}|x-\mathrm{a}| \sin \frac{1}{x-\mathrm{a}} \\ & =\lim _{\mathrm{h} \rightarrow 0}|\mathrm{a}-\mathrm{h}-\mathrm{a}| \sin \left(\frac{1}{\mathrm{a}-\mathrm{h}-\mathrm{a}}\right) \\ & =\lim _{\mathrm{h} \rightarrow 0}-\mathrm{h} \sin \frac{1}{\mathrm{~h}} \\ & =0 \times[\text { an oscillating number between }-1 \text { and } 1]=0\end{aligned}$

$\begin{aligned} & \text { R.H.L. }=\lim _{x \rightarrow a^{+}}|x-a| \sin \left(\frac{1}{x-a}\right) \\ & =\lim _{\mathrm{h} \rightarrow 0}|\mathrm{a}+\mathrm{h}-\mathrm{a}| \sin \left(\frac{1}{\mathrm{a}+\mathrm{h}-\mathrm{a}}\right) \\ & =\lim _{\mathrm{h} \rightarrow 0} \mathrm{~h} \sin \frac{1}{\mathrm{~h}} \\ & =0 \times[\text { an oscillating number between }-1 \text { and } \mathrm{l}]=0\end{aligned}$

Also $f(a)=0 \quad \ldots($ Given $)$

Thus L.H.L. $=$ R.H.L. $=\mathrm{f}(\mathrm{a})$

So, $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=\mathrm{a}$.

Question:8 Find which of the functions is continuous or discontinuous at the indicated points:
$f(x)=\left\{\begin{array}{cl} \frac{e^{\frac{1}{x}}}{1+e^{\frac{1}{x}}}, \text { if } & x \neq 0 \\ 0, & \text { if } x=0 \end{array}\right.$
at x = 0
Answer:

Given,
$f(x)=\left\{\begin{array}{cl} \frac{e^{\frac{1}{x}}}{1+e^{\frac{1}{x}}}, \text { if } & x \neq 0 \\ 0, & \text { if } x=0 \end{array}\right.$
We need to check its continuity at x = 0
A function f(x) is said to be continuous at x = c if,
Left-hand limit (LHL at x = c) = Right-hand limit(RHL at x = c) = f(c).
Mathematically, we can represent it as
$\lim _{h \rightarrow 0} f(c-h)=\lim _{h \rightarrow 0} f(c+h)=f(c)$
Where h is a very small number very close to 0 (h→0)
Now, according to the above theory-
f(x) is continuous at x = 4 if -

$\begin{aligned} &\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0} f(0+h)=f(0)\\ &\text { Clearly, }\\ &\lim _{h \rightarrow 0} f(-h)=\lim _{h \rightarrow 0}\left\{\frac{e^{\frac{1}{-h}}}{1+e^{-\frac{1}{-h}}}\right\}_{\{u \operatorname{sing} \text { equation } 1\}}\\ &\Rightarrow \mathrm{LHL}=\frac{\mathrm{e}^{\frac{1}{-0}}}{1+\mathrm{e}^{-\frac{1}{-0}}}=\frac{\mathrm{e}^{-\infty}}{1+\mathrm{e}^{-\infty}}=\frac{0}{1+0}=0\\ &\therefore \mathrm{LHL}=0 . . .(2) \end{aligned}$
Similarly, we proceed for RHL-
$\begin{array}{l} \lim _{\mathrm{h} \rightarrow 0} \mathrm{f}(\mathrm{h})=\lim _{\mathrm{h} \rightarrow 0}\left\{\frac{\mathrm{e}^{\frac{1}{\mathrm{~h}}}}{1+\mathrm{e}^{\frac{1}{\mathrm{~h}}}}\right\} {\{\text {using equation } 1\}} \\ \quad \quad \lim _{\mathrm{h} \rightarrow 0}\left\{\frac{\mathrm{e}^{\frac{1}{\mathrm{~h}}}}{\mathrm{e}^{\frac{1}{\mathrm{~h}}\left(1+\mathrm{e}^{-\frac{1}{\mathrm{~h}}}\right)}}\right\}=\lim _{\mathrm{h} \rightarrow 0}\left\{\frac{1}{\left(1+\mathrm{e}^{-\frac{1}{\mathrm{~h}}}\right)}\right\} \end{array}$
$\begin{array}{l} \Rightarrow \mathrm{RHL}=\frac{1}{1+\mathrm{e}-0}=\frac{1}{1+\mathrm{e}^{-\infty}}=\frac{1}{1+0}=1 \\ \therefore \mathrm{RHL}=1 \ldots(3) \end{array}$
And,
f(0) = 0 {using eqn 1} …(4)
From equations 2, 3, and 4, we can conclude that
$\lim _{h \rightarrow 0} \mathrm{f}(0-\mathrm{h}) \neq \lim _{h \rightarrow 0} \mathrm{f}(0+\mathrm{h})$
∴ f (x) is discontinuous at x = 0

Question:9

Find which of the functions is continuous or discontinuous at the indicated points:
$f(x)=\left\{\begin{array}{cc} \frac{x^{2}}{2}, \text { if } & 0 \leq x \leq 1 \\ 2 x^{2}-3 x+\frac{3}{2}, & \text { if } 1<x \leq 2 \end{array}\right.$

Answer:

We have, $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{ll}\frac{x^2}{2}, & \text { if } 0 \leq x \leq 1 \\ 2 x^2-3 x+\frac{3}{2}, & \text { if } 1<x \leq 2\end{array}\right.$ at $\mathrm{x}=1$

At $\mathrm{x}=1$

L.H.L. $=\lim _{x \rightarrow 1^{-}} \frac{x^2}{2}$

$\begin{aligned} & =\lim _{\mathrm{h} \rightarrow 0} \frac{(1-\mathrm{h})^2}{2} \\ & =\lim _{\mathrm{h} \rightarrow 0} \frac{1+\mathrm{h}^2-2 \mathrm{~h}}{2} \\ & =\frac{1}{2} \\ & \text { R.H.L. }=\lim _{x \rightarrow 1^{+}}\left(2 x^2-3 x+\frac{3}{2}\right) \\ & =\lim _{\mathrm{h} \rightarrow 0}\left[2(1+\mathrm{h})^2-3(1+\mathrm{h})+\frac{3}{2}\right] \\ & =2-3+\frac{3}{2} \\ & =\frac{1}{2}\end{aligned}$

Also $f(1)=\frac{1^2}{2}=\frac{1}{2}$ Thus L.H.L. $=$ R.H.L. $=f(1)$

Hence, $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=1$.

Question:10

Find which of the functions is continuous or discontinuous at the indicated points:
$f(x) = |x| + |x - 1|$ at x = 1

Answer:

We have, $\mathrm{f}(\mathrm{x})=|\mathrm{x}|+|\mathrm{x}-1|$ at $\mathrm{x}=1$ At $x=1$

$\begin{aligned} & \text { L.H.L. }=\lim _{x \rightarrow 1^{-}}[|x|+|x-1|] \\ & =\lim _{\mathrm{h}-? 0^{-}}[|1-\mathrm{h}|+|1-\mathrm{h}-1|] \\ & =1+0 \\ & =1\end{aligned}$

$\begin{aligned} & \text { And R.H.L. }=\lim _{x \rightarrow+}[|x|+x-1 \mid] \\ & =\lim _{\mathrm{h} \rightarrow 0}[|1+\mathrm{h}|+|1+\mathrm{h}-1|] \\ & =1+0 \\ & =1\end{aligned}$

Also $f(1)=|1|+|0|=1$

Thus, L.H.L. $=$ R.H.L $=f(1)$

Hence, $f(x)$ is continuous at $x=1$

Question:11

Find the value of k so that the function f is continuous at the indicated point:
$f(x)=\begin{array}{cl} 3 x-8, & \text { if } x \leq 5 \\ 2 k, & \text { if } x>5 \end{array} \text { at } x=5$

Answer:

Given,
$f(x)=\begin{array}{cl} 3 x-8, & \text { if } x \leq 5 \\ 2 k, & \text { if } x>5 \end{array} \text { at } x=5$
We need to find the value of k such that f(x) is continuous at x = 5.
A function f(x) is said to be continuous at x = c if,
Left-hand limit(LHL at x = c) = Right-hand limit(RHL at x = c) = f(c).
Mathematically, we can represent it as
$\lim _{h \rightarrow 0} f(c-h)=\lim _{h \rightarrow 0} f(c+h)=f(c)$
Where h is a very small number very close to 0 (h→0)
Now, let’s assume that f(x) is continuous at x = 5.
$\lim _{h \rightarrow 0} f(5-h)=\lim _{h \rightarrow 0} f(5+h)=f(5)$
As we have to find k so pick out a combination so that we get k in our equation.
In this question we take LHL = f(5)
$\\ \therefore \lim _{h \rightarrow 0} f(5-h)=f(5) \\ \Rightarrow \lim _{h \rightarrow 0}\{3(5-h)-8\}=2 k \\ \Rightarrow 3(5-0)-8=2 k \\ \Rightarrow 15-8=2 k \\ \Rightarrow 2 k=7 \\ \therefore k=7 / 2$

Question:12

Find the value of k so that the function f is continuous at the indicated point:
$f(x)=\frac{2^{x+2}-16}{4^{x}-16}, \quad$ if $x \neq 2$ if $x=2$ . at $x=2$$

Answer:

We have, $\mathrm{f}(\mathrm{x})= \begin{cases}\frac{2^{x+2}-16}{4^x-16}, & \text { if } x \neq 2 \\ \mathrm{k}, & \text { if } x=2\end{cases}$ Since, $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=2$

$\begin{aligned} & \therefore \mathrm{f}(2)=\lim _{x \rightarrow 2} \mathrm{f}(x) \\ & \therefore \mathrm{k}=\lim _{x \rightarrow 2} \frac{2^{x+2}-16}{4^x-16} \\ & =\lim _{x \rightarrow 2} \frac{4\left(2^x-4\right)}{\left(2^x-4\right)\left(2^x+4\right)} \\ & =\lim _{x \rightarrow 2} \frac{4}{2^x+4} \\ & =\frac{4}{4+4} \\ & =\frac{1}{2}\end{aligned}$

Question:13

Find the value of k so that the function f is continuous at the indicated point:
$\mathrm{f}(\mathrm{x})= \begin{cases}\frac{\sqrt{1+\mathrm{k} x}-\sqrt{1-\mathrm{k} x}}{x}, & \text { if }-1 \leq x<0 \\ \frac{2 x+1}{x-1}, & \text { if } 0 \leq x \leq 1\end{cases}$ at x= 0

Answer:

Given,

$\mathrm{f}(\mathrm{x})= \begin{cases}\frac{\sqrt{1+\mathrm{k} x}-\sqrt{1-\mathrm{k} x}}{x}, & \text { if }-1 \leq x<0 \\ \frac{2 x+1}{x-1}, & \text { if } 0 \leq x \leq 1\end{cases}$

We need to find the value of k such that f(x) is continuous at x = 0.
A function f(x) is said to be continuous at x = c if,
Left-hand limit(LHL at x = c) = Right-hand limit(RHL at x = c) = f(c).
Mathematically, we can represent it as
$\lim _{h \rightarrow 0} f(c-h)=\lim _{h \rightarrow 0} f(c+h)=f(c)$
Where h is a very small number very close to 0 (h→0)
Now, let’s assume that f(x) is continuous at x = 0.
$\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0} f(0+h)=f(0)$
Now, to find k, pick out a combination using which we get k in our equation.
In this question we take LHL = f(0)
$\begin{array}{l} \quad \lim _{h \rightarrow 0} f(-h)=f(0) \\ \Rightarrow \lim _{h \rightarrow 0}\left\{\frac{\sqrt{1+k(-h)}-\sqrt{1-k(-h)})}{-h}\right\}=\frac{2(0)+1}{(0)-1}\{\text { using eqn } 1\} \\ \Rightarrow \lim _{h \rightarrow 0}\left\{\frac{\sqrt{1+k h}-\sqrt{1-k h}}{h}\right\}=-1 \end{array}$
We can’t find the limit directly, because it is taking the 0/0 form.
Thus, we will rationalize it.
$\begin{aligned} &\Rightarrow \lim _{h \rightarrow 0}\left\{\frac{\sqrt{1+k h}-\sqrt{1-k h}}{h} \times \frac{\sqrt{1+k h}+\sqrt{1-k h}}{\sqrt{1+k h}+\sqrt{1-k h}}\right\}=-1\\ &\text { Using }(a+b)(a-b)=a^{2}-b^{2}, \text { we have }-\\ &\lim _{h \rightarrow 0}\left\{\frac{(\sqrt{1+k h})^{2}-(\sqrt{1-k h)})^{2}}{h(\sqrt{1+k h}+\sqrt{1-k h})}\right\}=-1\\ &\Rightarrow \lim _{h \rightarrow 0}\left\{\frac{2 k h}{h(\sqrt{1+k h}+\sqrt{1-k h})}\right\}=-1\\ &\lim _{h \rightarrow 0}\left\{\frac{2 k}{(\sqrt{1+k h}+\sqrt{1-k h})}\right\}=\\ &\Rightarrow \frac{2 k}{\sqrt{1+k(0)}+\sqrt{1-k(0)}}=-1\\ &\therefore 2 k / 2=-1 \\&\therefore k =-1 \end{aligned}$

Question:14

Find the value of k so that the function f is continuous at the indicated point:
$\begin{array}{c} f(x)=\frac{1-\cos \mathrm{kx}}{\mathrm{x} \sin \mathrm{x}}, \quad \text { if } \mathrm{x} \neq 0 \\ \frac{1}{2}, \quad \text { if } \mathrm{x}=0 \end{array}$ at x= 0

Answer:

Given,
$\begin{array}{c} f(x)=\frac{1-\cos \mathrm{kx}}{\mathrm{x} \sin \mathrm{x}}, \quad \text { if } \mathrm{x} \neq 0 \\ \frac{1}{2}, \quad \text { if } \mathrm{x}=0 \end{array}$
We need to find the value of k such that f(x) is continuous at x = 0.
A function f(x) is said to be continuous at x = c if,
Left-hand limit(LHL at x = c) = Right-hand limit(RHL at x = c) = f(c).
Mathematically, we can represent it as
$\lim _{h \rightarrow 0} f(c-h)=\lim _{h \rightarrow 0} f(c+h)=f(c)$
Where h is a very small number very close to 0 (h→0)
Now, let’s assume that f(x) is continuous at x = 0.
$\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0} f(0+h)=f(0)$
to find k we have to pick out a combination so that we get k in our equation.
In this question we take LHL = f(0)
$\begin{array}{l} \lim _{h \rightarrow 0} f(-h)=f(0) \\ \quad \lim _{h \rightarrow 0}\left\{\frac{1-\cos k(-h)}{(-h) \sin (-h)}\right\}=\frac{1}{2}\{u \text { ing equation } 1\} \end{array}$
$\begin{array}{l} \because \cos (-x)=\cos x \text { and } \sin (-x)=-\sin x \\ \quad \lim _{h \rightarrow 0}\left\{\frac{1-\cos k(h)}{(h) \sin (h)}\right\}=\frac{1}{2} \\ \text { Also, } 1-\cos x=2 \sin ^{2}(x / 2) \\ \quad \lim _{h \rightarrow 0}\left\{\frac{2 \sin ^{2}\left(\frac{k h}{2}\right)}{(h) \sin (h)}\right\}=\frac{1}{2} \end{array}$
This limit can be evaluated directly by putting the value of h because it is taking an indeterminate form (0/0)
Thus, we use the sandwich or squeeze theorem according to which -
$\lim _{x \rightarrow 0} \frac{\sin x}{x}=1 \\ \Rightarrow {\lim_{h\rightarrow 0} }\left\{\frac{\sin ^{2}\left(\frac{k h}{2}\right)}{(h) \sin (h)}\right\}=\frac{1}{4}$
Dividing and multiplying by $(kh/2)^2$ to match the form in the formula we have-
$\begin{aligned} &\lim _{h \rightarrow 0}\left\{\frac{\sin ^{2}\left(\frac{\mathrm{kh}}{2}\right)}{(\mathrm{h}) \sin (\mathrm{h}) \times\left(\frac{\mathrm{kh}}{2}\right)^{2}} \times\left(\frac{\mathrm{kh}}{2}\right)^{2}\right\}=\frac{1}{4}\\ &\text { Using algebra of limits we get - }\\ &\lim _{h \rightarrow 0}\left(\frac{\sin \frac{k h}{2}}{\frac{k h}{2}}\right)^{2} \times \lim _{h \rightarrow 0} \frac{k^{2}}{4}\left(\frac{h}{\sin h}\right)=\frac{1}{4} \end{aligned}$
$\begin{aligned} &\text { Applying the formula- }\\ &\Rightarrow 1 \times\left(\mathrm{k}^{2} / 4\right)=(1 / 4)\\ &\Rightarrow \mathrm{k}^{2}=1\\ &\Rightarrow(k+1)(k-1)=0\\ &\therefore \mathrm{k}=1 \text { or } \mathrm{k}=-1 \end{aligned}$

Question:15

Prove that the function f defined by

$f(x)=\left\{\begin{array}{cl} \frac{x}{|x|+2 x^{2}}, & x \neq 0 \\ k, & x=0 \end{array}\right.$
remains discontinuous at x=0, regardless of the choice of k.

Answer:

we have $\mathrm{f}(\mathrm{x})= \begin{cases}\frac{x}{|x|+2 x^2}, & x \neq 0 \\ \mathrm{k} & x=0\end{cases}$

At $x=0$

$\begin{aligned} & \text { L.H.L. }=\lim _{x \rightarrow 0^{+}} \frac{(0-\mathrm{h})}{|0-\mathrm{h}|+2(0-\mathrm{h})^2} \\ & =\lim _{\mathrm{h} \rightarrow 0} \frac{-\mathrm{h}}{\mathrm{h}+2 \mathrm{~h}^2} \\ & =\lim _{\mathrm{h} \rightarrow 0} \frac{-1}{1+2 \mathrm{~h}} \\ & =-1 \\ & \text { R.H.L. }=\lim _{x \rightarrow 0^{+}} \frac{x}{|x|+2 x^2} \\ & =\lim _{\mathrm{h} \rightarrow 0} \frac{0+\mathrm{h}}{|0+\mathrm{h}|+2(0+\mathrm{h})^2} \\ & =\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{~h}}{\mathrm{~h}+2 \mathrm{~h}^2} \\ & =\lim _{\mathrm{h} \rightarrow 0} \frac{1}{1+2 \mathrm{~h}} \\ & =1\end{aligned}$

Since, L.H.L. $\neq$ R.H.L. for any value of $k$.

Hence, $f(x)$ is discontinuous at $x=0$ regardless of the choice of $k$.

Question:16

Find the values of a and b such that the function f defined by
$f(x)=\left\{\begin{array}{ll} \frac{x-4}{|x-4|}+a, & \text { if } x<4 \\ a+b &, \text { if } x=4 \\ \frac{x-4}{|x-4|}+b, & \text { if } x>4 \end{array}\right.$
is a continuous function at x = 4.

Answer:

Given,
$f(x)=\left\{\begin{array}{ll} \frac{x-4}{|x-4|}+a, & \text { if } x<4 \\ a+b & , \text { if } x=4 \\ \frac{x-4}{|x-4|}+b, & \text { if } x>4 \end{array}\right.$ …(1)
We need to find the value of a & b such that f(x) is continuous at x = 4.
A function f(x) is said to be continuous at x = c if,
Left-hand limit(LHL at x = c) = Right-hand limit(RHL at x = c) = f(c).
Mathematically we can represent it as
$\lim _{h \rightarrow 0} f(c-h)=\lim _{h \rightarrow 0} f(c+h)=f(c)$
Where h is a very small number very close to 0 (h→0)
Now, let’s assume that f(x) is continuous at x = 4.
$\therefore \lim _{h \rightarrow 0} \mathrm{f}(4-\mathrm{h})=\lim _{h \rightarrow 0} \mathrm{f}(4+\mathrm{h})=\mathrm{f}(4)$
to find a & b, we have to pick out a combination so that we get a or b in our equation.
In this question first, we take LHL = f(4)
$\therefore \lim _{h \rightarrow 0} f(4-h)=f(4)$
$\Rightarrow \lim _{h \rightarrow 0}\left\{\frac{4-h-4}{|4-h-4|}+a\right\}=a+b$ {using equation 1}
$\Rightarrow \lim _{h \rightarrow 0}\left\{\frac{-h}{|-h|}+a\right\}=a+b$
$\because$ h > 0 as defined in the theory above.
$\therefore|-h|=h$
$\Rightarrow \lim _{h \rightarrow 0}\left\{\frac{-h}{h}+a\right\}=a+b$
$\Rightarrow \lim _{h \rightarrow 0}\{a-1\}=a+b$
$\Rightarrow$ a - 1 = a + b
$\therefore$ b = -1
Now, taking another combination,
RHL = f(4)
$\lim _{h \rightarrow 0} \mathrm{f}(4+\mathrm{h})=\mathrm{f}(4)$
$\Rightarrow \lim _{h \rightarrow 0}\left\{\frac{4+h-4}{|4+h-4|}+b\right\}=a+b$ {using equation 1}
$\underset{h \rightarrow 0}{\lim }\left\{\frac{h}{|h|}+b\right\}=a+b$
$\because$ h > 0 as defined in the theory above.
$\therefore$ |h| = h
$\Rightarrow \lim _{h \rightarrow 0}\left\{\frac{h}{h}+b\right\}=a+b$
$\Rightarrow \lim _{h \rightarrow 0}\{b+1\}=a+b$
⇒ b + 1 = a + b
∴ a = 1
Hence,
a = 1 and b = -1

Question:17

Given the function $f(x)=\frac{1}{x+2}$. Find the point of discontinuity of the composite function y = f(f(x)).

Answer:

Given,
$F(x)=\frac{1}{x+2}$
we have to find the points discontinuity of composite function f(f(x))

As f(x) is not defined at x = -2 as denominator becomes 0, at x = -2.

$\therefore$ x = -2 is a point of discontinuity
$\because f(f(x))=f\left(\frac{1}{x+2}\right)=\frac{1}{\frac{1}{x+2}+2}=\frac{x+2}{2 x+5}$
And f(f(x)) is not defined at x = -5/2 as the denominator becomes 0, at x = -5/2.
∴ x = -5/2 is another point of discontinuity
Thus f (f(x)) has 2 points of discontinuity at x = -2 and x = -5/2

Question:18

Find all points of discontinuity of the function $f(t)=\frac{1}{t^{2}+t-2}, \quad t=\frac{1}{x-1}$.

Answer:

We have, $\mathrm{f}(\mathrm{t})=\frac{1}{\mathrm{t}^2+\mathrm{t}-2}$

Where $\mathrm{t}=\frac{1}{x-1}$

$\begin{aligned} & \therefore \mathrm{f}(\mathrm{t})=\frac{1}{\left(\frac{1}{x-1}\right)^2+\frac{1}{x-1}-2} \\ & =\frac{(x-1)^2}{1+(x-1)-2(x-1)^2} \\ & =\frac{(x-1)^2}{-\left(2 x^2-5 x+2\right)} \\ & =\frac{(x-1)^2}{(2 x-1)(2-x)}\end{aligned}$

So, $\mathrm{f}(\mathrm{t})$ is discontinuous at $2 \mathrm{x}-1=0$

$\begin{aligned} & \Rightarrow \mathrm{x}=\frac{1}{2} \text { and } 2-\mathrm{x}=0 \\ & \Rightarrow \mathrm{x}=2\end{aligned}$

Also $\mathrm{f}(\mathrm{t})$ is discontinuous at $\mathrm{x}=1$, where $\mathrm{t}=\frac{1}{x-1}$ is discontinuous.

Question:19

Show that the function f(x) = |sin x + cos x| is continuous at x = $\pi$.

Answer:

Given,

$f(x)=|\sin x+\cos x| \underline{\ldots}(1)$

We need to prove that f(x) is continuous at x = π

A function f(x) is said to be continuous at x = c if,

Left-hand limit(LHL at x = c) = Right-hand limit(RHL at x = c) = f(c).

Mathematically we can represent it as

$\lim _{h \rightarrow 0} f(c-h)=\lim _{h \rightarrow 0} f(c+h)=f(c)$

Where h is a very small number very close to 0 (h→0)

Now according to the above theory-

f(x) is continuous at x = π if -

$\lim _{h \rightarrow 0} f(\pi-h)=\lim _{h \rightarrow 0} f(\pi+h)=f(\pi)$

Now,

LHL = $\lim _{h \rightarrow 0} f(\pi-h)$

⇒ LHL = $\lim _{h \rightarrow 0}\{|\sin (\pi-h)+\cos (\pi-h)|\}$ {using eqn 1}

$\because \sin (\pi-x)=\sin x \& \cos (\pi-x)=-\cos x$

$\Rightarrow \mathrm{LHL}=\lim _{\mathrm{h} \rightarrow 0}|\sinh -\cosh |$

$\Rightarrow \mathrm{LHL}=|\sin 0-\cos 0|=|0-1|$

$\therefore \mathrm{LHL}=1 \underline{\ldots(2)}$

Similarly, we proceed for RHL-

$\operatorname{RHL}=\lim _{h \rightarrow 0} f(\pi+h)$

$\Rightarrow \mathrm{RHL}=\lim _{\mathrm{h} \rightarrow 0}\{|\sin (\pi+\mathrm{h})+\cos (\pi+\mathrm{h})|\}$ {using eqn 1}

$\because \sin (\pi+x)=-\sin x \& \cos (\pi+x)=-\cos x$

$\Rightarrow \mathrm{RHL}=\lim _{\mathrm{h} \rightarrow 0}|-\sin \mathrm{h}-\cosh |$

$\Rightarrow \mathrm{RHL}=|-\sin 0-\cos 0|=|0-1|$

$\therefore \mathrm{RHL}=1 \ldots(3)$

$\text { Also, } f(\pi)=|\sin \pi+\cos \pi|=|0-1|=1 \underline{\ldots(4)}$

Now from equations 2, 3, and 4, we can conclude that

$\lim _{h \rightarrow 0} \mathrm{f}(\pi-\mathrm{h})=\lim _{h \rightarrow 0} \mathrm{f}(\pi+\mathrm{h})=\mathrm{f}(\pi)=1$

∴ f(x) is continuous at x = π is proved

Question:20

Examine the differentiability of f, where f is defined by

$f(x)=\left\{\begin{array}{cc} x[x], & \text { if } 0 \leq x<2 \\ (x-1) x, & \text { if } 2 \leq x<3 \end{array}\right. \text { at } x=2$.

Answer:

Given,
$f(x)=\left\{\begin{array}{cc} x[x], & \text { if } 0 \leq x<2 \\ (x-1) x, & \text { if } 2 \leq x<3 \end{array}\right. \text { at } x=2$ …(1)
We need to check whether f(x) is continuous and differentiable at x = 2
A function f(x) is said to be continuous at x = c if,
Left-hand limit(LHL at x = c) = Right-hand limit(RHL at x = c) = f(c).
Mathematically we can represent it as
$\lim _{h \rightarrow 0} f(c-h)=\lim _{h \rightarrow 0} f(c+h)=f(c)$
Where h is a very small number very close to 0 (h→0)
And a function is said to be differentiable at x = c if it is continuous there and
Left-hand derivative (LHD at x = c) = Right-hand derivative (RHD at x = c) = f(c).
Mathematically we can represent it as
$\lim _{x \rightarrow c^{-}} \frac{f(x)-f(c)}{x-c}=\lim _{x \rightarrow c^{+}} \frac{f(x)-f(c)}{x-c}$
$\lim _{h \rightarrow 0} \frac{f(c-h)-f(c)}{c-h-c}=\lim _{h \rightarrow 0} \frac{f(c+h)-f(c)}{c+h-c}$
Finally, we can state that for a function to be differentiable at x = c

$\lim _{h \rightarrow 0} \frac{f(c-h)-f(c)}{-h}=\lim _{h \rightarrow 0} \frac{f(c+h)-f(c)}{h}$
Checking for the continuity:
Now according to the above theory-
f(x) is continuous at x = 2 if -
$\lim _{h \rightarrow 0} f(2-h)=\lim _{h \rightarrow 0} f(2+h)=f(2)$
$\therefore \mathrm{LHL}=\lim _{\mathrm{h} \rightarrow 0} \mathrm{f}(2-\mathrm{h})$
$\Rightarrow \mathrm{LHL}=\lim _{\mathrm{h} \rightarrow 0}(2-\mathrm{h})[2-\mathrm{h}]_{\{\text {using equation } 1\}}$
Note: As [.] represents the greatest integer function which gives the greatest integer less than the number inside [.].
E.g. [1.29] = 1; [-4.65] = -5 ; [9] = 9
[2-h] is just less than 2 say 1.9999 so [1.999] = 1
⇒ LHL = (2-0) ×1
∴ LHL = 2 …(2)
Similarly,
RHL = $\lim _{h \rightarrow 0} f(2+h)$
⇒ RHL = $\lim _{h \rightarrow 0}(2+h-1)(2+h)_{\{\text {using equation } 1\}}$
$\therefore \mathrm{RHL}=(1+0)(2+0)=2 \ldots(3)$
And, f(2) = (2-1)(2) = 2 …(4) {using equation 1}
From equation 2,3 and 4, we observe that:
$\lim _{h \rightarrow 0} f(2-h)=\lim _{h \rightarrow 0} f(2+h)=f(2)=2$
∴ f(x) is continuous at x = 2. So we will proceed now to check the differentiability.
Checking for the differentiability:
Now according to the above theory-
f(x) is differentiable at x = 2 if -
$\lim _{h \rightarrow 0} \frac{f(2-h)-f(2)}{-h}=\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$
$\therefore$ LHD = $\lim _{h \rightarrow 0} \frac{f(2-h)-f(2)}{-h}$
⇒ LHD = $\lim _{h \rightarrow 0} \frac{(2-h)[2-h]-(2-1)(2)}{-h} \quad\{\text { using equation } 1\}$
Note: As [.] represents the greatest integer function which gives the greatest integer less than the number inside [.].
E.g. [1.29] = 1 ; [-4.65] = -5 ; [9] = 9
[2-h] is just less than 2 say 1.9999 so [1.999] = 1

⇒ LHD = $\lim _{h \rightarrow 0} \frac{(2-h) \times 1-2}{-h}$
⇒ LHD = $\lim _{h \rightarrow 0} \frac{-h}{-h}=\lim _{h \rightarrow 0} 1$
$\therefore \mathrm{LHD}=1 \underline{\ldots}(5)$
Now,
RHD = $\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}$
⇒ RHD = $\lim _{h \rightarrow 0} \frac{(2+h-1)(2+h)-(2-1)(2)}{h} \quad\{\text { using equation } 1\}$
⇒ RHD = $\lim _{h \rightarrow 0} \frac{(1+h)(2+h)-2}{h}=\lim _{h \rightarrow 0} \frac{2+h^{2}+3 h-2}{h}$
∴ RHD = $\lim _{h \rightarrow 0} \frac{h(h+3)}{h}=\lim _{h \rightarrow 0}(h+3)$
$\Rightarrow \mathrm{RHD}=0+3=3 \underline{\ldots}(6)$
Clearly from equations 5 and 6, we can conclude that-
(LHD at x=2) ≠ (RHD at x = 2)
∴ f(x) is not differentiable at x = 2

Question:21

Examine the differentiability of f, where f is defined by
$f(x)=\left\{\begin{array}{cl} x^{2} \sin \frac{1}{x}, & \text { if } x \neq 0 \\ 0, & \text { if } x=0 \end{array}\right. \text { at } x=0$

Answer:

Given,
$f(x)=\left\{\begin{array}{cl} x^{2} \sin \frac{1}{x}, & \text { if } x \neq 0 \\ 0, & \text { if } x=0 \end{array}\right. \text { at } x=0$
We need to check whether f(x) is continuous and differentiable at x = 0
A function f(x) is said to be continuous at x = c if,
Left-hand limit(LHL at x = c) = Right-hand limit(RHL at x = c) = f(c).
Mathematically we can represent it as
$\lim _{h \rightarrow 0} f(c-h)=\lim _{h \rightarrow 0} f(c+h)=f(c)$
Where h is a very small number very close to 0 (h→0)
And a function is said to be differentiable at x = c if it is continuous there and
Left-hand derivative(LHD at x = c) = Right-hand derivative(RHD at x = c) = f(c).
Mathematically we can represent it as
$\lim _{x \rightarrow c^{-}} \frac{f(x)-f(c)}{x-c}=\lim _{x \rightarrow c^{+}} \frac{f(x)-f(c)}{x-c}$
$\lim _{h \rightarrow 0} \frac{f(c-h)-f(c)}{c-h-c}=\lim _{h \rightarrow 0} \frac{f(c+h)-f(c)}{c+h-c}$
Finally, we can state that for a function to be differentiable at x = c
$\lim _{h \rightarrow 0} \frac{f(c-h)-f(c)}{-h}=\lim _{h \rightarrow 0} \frac{f(c+h)-f(c)}{h}$
Checking for the continuity:
Now according to the above theory-
f(x) is continuous at x = 0 if -
$\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0} f(0+h)=f(0) \\ \therefore L H L=h \rightarrow 0 \\ \Rightarrow L H L=\lim _{h \rightarrow 0}\left\{(-h)^{2} \sin \left(\frac{1}{-h}\right)\right\}_{\{u \operatorname{sing}} \text { equation } \left.1\right\}$
As sin (-1/h) is going to be some finite value from -1 to 1 as h→0
∴ LHL = $0^2$ × (finite value) = 0
∴ LHL = 0 …(2)
Similarly,
$\lim _{\mathrm{RHL}}=\lim _{h \rightarrow 0} \mathrm{f}(\mathrm{h}) \\ \Rightarrow \mathrm{RHL}=\lim _{\mathrm{h} \rightarrow 0}\left\{(\mathrm{~h})^{2} \sin \left(\frac{1}{\mathrm{~h}}\right)\right\}_{\{\text {using equation } 1\}}$
As sin (1/h) is going to be some finite value from -1 to 1 as h→0
∴ RHL = $0^2$(finite value) = 0 …(3)
And, f(0) = 0 {using equation 1} …(4)
From equation 2,3 and 4, we observe that:
$\lim _{h \rightarrow 0} f(0-h)=\lim _{h \rightarrow 0} f(0+h)=f(0)$
∴ f(x) is continuous at x = 0. So we will proceed now to check the differentiability.
Checking for the differentiability:
Now according to the above theory-
f(x) is differentiable at x = 0 if -
$\\ \lim _{h \rightarrow 0} \frac{f(0-h)-f(0)}{-h}=\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h} \\ \therefore L H D=\lim_{h \rightarrow 0 }\frac{f(-h)-f(0)}{-h} \\ \Rightarrow L H D=\lim_{h \rightarrow 0 }\frac{(-h)^{2} \sin \left(\frac{1}{-h}\right)-0}{-h} \quad\{\text { using equation } 1\} \\ \Rightarrow L H D=\lim _{h \rightarrow 0} \frac{h^{2} \sin \left(\frac{1}{-h}\right)}{-h}=\lim _{h \rightarrow 0}\left\{h \sin \left(\frac{1}{h}\right)\right\}$
As sin (1/h) is going to be some finite value from -1 to 1 as h→0
∴ LHD = 0×(some finite value) = 0
∴ LHD = 0 …(5)
Now,
$\operatorname{RHD}=\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h} \\ \Rightarrow R H D=h \rightarrow 0 \frac{(h)^{2} \sin \left(\frac{1}{h}\right)-0}{h} \quad\{\text { using equation } 1\} \\$
$\Rightarrow \mathrm{RHD}=\lim _{\mathrm{h} \rightarrow 0}\left\{\mathrm{~h} \sin \left(\frac{1}{\mathrm{~h}}\right)\right\}$
As sin (1/h) is going to be some finite value from -1 to 1 as h→0
∴ RHD = 0×(some finite value) = 0
∴ RHD = 0 …(6)
Clearly from equations 5 and 6, we can conclude that-
(LHD at x=0) = (RHD at x = 0)
∴ f(x) is differentiable at x = 0

Question:22

Examine the differentiability of f, where f is defined by
$f(x)=\left\{\begin{array}{ll} 1+x, & \text { if } x \leq 2 \\ 5-x, & \text { if } x>2 \end{array}\right. \text { at } x=2$

Answer:

Given,
$f(x)=\left\{\begin{array}{ll} 1+x, & \text { if } x \leq 2 \\ 5-x, & \text { if } x>2 \end{array}\right. \text { at } x=2$
We need to check whether f(x) is continuous and differentiable at x = 2
A function f(x) is said to be continuous at x = c if,
Left-hand limit(LHL at x = c) = Right-hand limit(RHL at x = c) = f(c).
Mathematically we can represent it as
$\lim _{h \rightarrow 0} f(c-h)=\lim _{h \rightarrow 0} f(c+h)=f(c)$.
Where h is a very small number very close to 0 (h→0)
And a function is said to be differentiable at x = c if it is continuous there and
LLeft-handderivative(LHD at x = c) = Right-hand derivative(RHD at x = c) = f(c).
Mathematically we can represent it as
$\\ \lim _{x \rightarrow c^{-}} \frac{f(x)-f(c)}{x-c}=\lim _{x \rightarrow c^{+}} \frac{f(x)-f(c)}{x-c} \\ \lim _{h \rightarrow 0} \frac{f(c-h)-f(c)}{c-h-c}=\lim _{h \rightarrow 0} \frac{f(c+h)-f(c)}{c+h-c}$
Finally, we can state that for a function to be differentiable at x = c
$\lim _{h \rightarrow 0} \frac{f(c-h)-f(c)}{-h}=\lim _{h \rightarrow 0} \frac{f(c+h)-f(c)}{h}$
Checking for the continuity:
Now according to the above theory-
f(x) is continuous at x = 2 if -
$\lim _{h \rightarrow 0} f(2-h)=\lim _{h \rightarrow 0} f(2+h)=f(2) \\ \therefore L H L=\lim _{h \rightarrow 0} f(2-h)$
$\begin{aligned} &\Rightarrow \mathrm{LHL}=\lim _{\mathrm{h} \rightarrow 0}\{1+(2-\mathrm{h})\}_{\{\text {using equation } 1\}}\\ &\Rightarrow \mathrm{LHL}=\lim _{\mathrm{h} \rightarrow 0}\{3-\mathrm{h}\}\\ &\therefore L H L=(3-h)=3\\ &\therefore \mathrm{LHL}=3 \ldots(2)\\ &\text { Similarly, }\\ &\lim _{\mathrm{RHL}}=\operatorname{h}_{h \rightarrow 0} \mathrm{f}(2+\mathrm{h})\\ &\Rightarrow \mathrm{RHL}=\lim _{\mathrm{h} \rightarrow 0}\{5-(2+\mathrm{h})\}_{\{\text {using equation } 1\}}\\ &\Rightarrow \mathrm{RHL}=\lim _{h \rightarrow 0}\{3+\mathrm{h}\}\\ &\therefore \mathrm{RHL}=3+0=3 . .0(3) \end{aligned}$
And, f(2) = 1 + 2 = 3 {using equation 1} …(4)
From equation 2,3 and 4, we observe that:
$\lim _{h \rightarrow 0} \mathrm{f}(2-\mathrm{h})=\lim _{h \rightarrow 0} \mathrm{f}(2+\mathrm{h})=\mathrm{f}(2)=3$
∴ f(x) is continuous at x = 2. So we will proceed now to check the differentiability.
Checking for the differentiability:
Now according to the above theory-
f(x) is differentiable at x = 2 if -
$\begin{array}{l} \lim _{h \rightarrow 0} \frac{f(2-h)-f(2)}{-h}=\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h} \\ \therefore L H D=\lim _{h \rightarrow 0} \frac{f(2-h)-f(2)}{-h} \end{array}$
$\begin{aligned} &\Rightarrow \mathrm{LHD}=\lim _{h \rightarrow 0} \frac{1+(2-\mathrm{h})-(1+2)}{-\mathrm{h}} \quad\{\text { using equation } 1\}\\ &\Rightarrow \mathrm{LHD}=\lim _{\mathrm{h} \rightarrow 0} \frac{3-\mathrm{h}-3}{-\mathrm{h}}=\lim _{\mathrm{h} \rightarrow 0} 1\\ &\therefore \mathrm{LHD}=1 . .0(5)\\ &\text { Now, }\\ &\lim _{\mathrm{RHD}}=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{f}(2+\mathrm{h})-\mathrm{f}(2)}{\mathrm{h}}\\ &\Rightarrow \mathrm{RHD}=\lim _{\mathrm{h} \rightarrow 0} \frac{5-(2+\mathrm{h})-3}{\mathrm{~h}} \quad\{\text { using equation } 1\}\\ &\Rightarrow \lim _{\mathrm{RHD}}=\underset{\mathrm{h} \rightarrow 0}{-\mathrm{h}}=\lim _{\mathrm{h} \rightarrow 0}-1\\ &\therefore \mathrm{RHD}=-1 \ldots(6) \end{aligned}$
Clearly from equations 5 and 6, we can conclude that-
(LHD at x=2) ≠ (RHD at x = 2)
∴ f(x) is not differentiable at x = 2

Question:23

Show that f(x) = |x - 5| is continuous but not differentiable at x = 5.

Answer:

We have $f(x)=|x-5|$

$\Rightarrow \mathrm{f}(\mathrm{x})= \begin{cases}-(x-5), & \text { if } x-5<0 \text { or } x<5 \\ x-5, & \text { if } x-5>0 \text { or } x>5\end{cases}$

$\begin{aligned} & \text { L.H.L. } \lim _{h \rightarrow 5^{-}} f(x)=-(x-5) \\ & =\lim _{h \rightarrow 0}-(5-h-5) \\ & =\lim _{h \rightarrow 0} h=0 \\ & \text { R.H.L. } \lim _{x \rightarrow 5^{+}} f(x)=x-5 \\ & =\lim _{h \rightarrow 0}(5+h-5) \\ & =\lim _{h \rightarrow 0} h=0\end{aligned}$

L.H.L. = R.H.L.

So, $f(x)$ is continuous at $x=5$

Now, for differentiability

$\begin{aligned} & \operatorname{Lf}^{\prime}(5)=\lim _{h \rightarrow 0} \frac{f(5-h)-f(5)}{-h} \\ & =\lim _{h \rightarrow 0} \frac{-(5-h-5)-(5-5)}{-h} \\ & =\lim _{h \rightarrow 0} \frac{h}{-h} \\ & =-1 \\ & \operatorname{Rf}^{\prime}(5)=\lim _{h \rightarrow 0} \frac{f(5+h)-f(5)}{h} \\ & =\lim _{h \rightarrow 0} \frac{(5+h-5)-(5-5)}{h} \\ & =\lim _{h \rightarrow 0} \frac{h-0}{h} \\ & =1 \\ & \because \operatorname{Lf}^{\prime}(5) \neq \operatorname{Rd}(5)\end{aligned}$

Hence, $\mathrm{f}(\mathrm{x})$ is not differentiable at $\mathrm{x}=5$.

Question:24 A function $f: R\rightarrow R$ satisfies the equation f(x + y) = f(x) f(y) for all x, y$\in R,f(x)\neq 0$. Suppose that the function is differentiable at x = 0 and f’(0) = 2. Prove that f’(x) = 2f(x).

Answer:

Given that, $f: R \rightarrow R$ satisfies the equation $f(x+y)=f(x) f(y)$ for all $x, y \in R, f(x) \neq 0$.

Let us take any point $\mathrm{x}=0$ at which the function $\mathrm{f}(\mathrm{x})$ is differentiable.

$\begin{aligned} & \therefore \mathrm{f}^{\prime}(0)=\lim _{h \rightarrow 0} \frac{\mathrm{f}(0+\mathrm{h})-\mathrm{f}(0)}{\mathrm{h}} \\ & 2=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{f}(0) \cdot \mathrm{f}(\mathrm{h})-\mathrm{f}(0)}{\mathrm{h}} . \\ & \Rightarrow 2=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{f}(0)[\mathrm{f}(\mathrm{h})-1]}{\mathrm{h}}\end{aligned}$

Now $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$

$\begin{aligned} & =\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{f}(x) \cdot \mathrm{f}(\mathrm{h})-\mathrm{f}(x)}{\mathrm{h}} \\ & =\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{f}(x)[\mathrm{f}(\mathrm{h})-1]}{\mathrm{h}} \\ & =2 \mathrm{f}(\mathrm{x})\end{aligned}$

Hence, $f^{\prime}(x)=2 f(x)$.

Question:25

Differentiate each of the following w.r.t. x
$2 ^{\cos ^2 x}$

Answer:

Given: $2 ^{\cos ^2 x}$
Let Assume $y=2 ^{\cos ^2 x}$
Now, Taking Log on both sides we get,
$\begin{aligned} &\log y=\log 2^{\cos ^{2} x}\\ &\log \mathrm{y}=\cos ^{2} \mathrm{x} \cdot \log 2\\ &\text { Now, Differentiate w.r.t } x\\ &\frac{1}{y} \frac{d y}{d x}=\frac{d}{d x}\left[\cos ^{2} x \cdot \log 2\right]\\ &\frac{1}{y} \frac{d y}{d x}=[2 \cos x \cdot(-\sin x) \cdot \log 2]\\ &\frac{d y}{d x}=y[2 \cos x \cdot(-\sin x) \cdot \log 2] \end{aligned}$
Now, substitute the value of y
$\\ \frac{\mathrm{dy}}{\mathrm{dx}}=-2^{\cos ^{2} x}[2 \cos \mathrm{x} \cdot(\sin \mathrm{x}) \cdot \log 2] \\ $

$\text { Hence, } \frac{\mathrm{dy}}{\mathrm{dx}}=-2^{\cos ^{2} x}[2 \cos \mathrm{x} \cdot(\sin \mathrm{x}) \cdot \log 2]$

Question:26 Differentiate each of the following w.r.t. x

$\frac{8^x}{x^8}$

Answer:

Let $\mathrm{y}=\frac{8^x}{x^8}$

Taking $\log$ on both sides, we get,

$\begin{aligned} & \log \mathrm{y}=\log \frac{8^x}{x^8} \\ & \Rightarrow \log \mathrm{y}=\log 8^x-\log x^8 \\ & \Rightarrow \log \mathrm{y}=\mathrm{x} \log 8-8 \log \mathrm{x}\end{aligned}$

Differentiating both sides w.r.t. X

$\begin{aligned} & \Rightarrow \frac{1}{y} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}=\log 8.1-\frac{8}{x} \\ & \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=y\left[\log 8-\frac{8}{x}\right]\end{aligned}$

Hence, $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{8^x}{x^8}\left[\log 8-\frac{8}{x}\right]$

Question:27 Differentiate each of the following w.r.t. x
$\log\left ( x+\sqrt{x^2 +a}\right )$

Answer:

Let $\mathrm{y}=\log \left(x+\sqrt{x^2+\mathrm{a}}\right)$

Differentiating both sides w.r.t. x

$\begin{aligned} & \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}} \log \left(x+\sqrt{x^2+\mathrm{a}}\right) \\ & =\frac{1}{x+\sqrt{x^2+\mathrm{a}}} \cdot \frac{\mathrm{d}}{\mathrm{dx}}\left(x+\sqrt{x^2+\mathrm{a}}\right) \\ & =\frac{1}{x+\sqrt{x^2+\mathrm{a}}} \cdot\left[1+\frac{1}{2 \sqrt{x^2+\mathrm{a}}} \times \frac{\mathrm{d}}{\mathrm{dx}}\left(x^2+\mathrm{a}\right)\right] \\ & =\frac{1}{x+\sqrt{x^2+\mathrm{a}}} \cdot\left[1+\frac{1}{2 \sqrt{x^2+\mathrm{a}}} \cdot 2 x\right] \\ & =\frac{1}{x+\sqrt{x^2+\mathrm{a}}} \cdot\left[1+\frac{x}{\sqrt{x^2+\mathrm{a}}}\right] \\ & =\frac{1}{x+\sqrt{x^2+\mathrm{a}}} \cdot\left(\frac{\sqrt{x^2+\mathrm{a}+x}}{\sqrt{x^2+\mathrm{a}}}\right) \\ & =\frac{1}{\sqrt{x^2+\mathrm{a}}}\end{aligned}$

Hence. $\frac{d y}{d x}=\frac{1}{\sqrt{x^2+a}}$

Question:28

Differentiate each of the following w.r.t. x
$\log [\log (\log x^5)]$

Answer:

Let $\mathrm{y}=\log \left[\log \left(\log x^5\right)\right]$

Differentiating both sides w.r.t. x

$\begin{aligned} & \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}} \log \left[\log \left(\log x^5\right)\right] \\ & =\frac{1}{\log \left(\log x^5\right)} \times \frac{\mathrm{d}}{\mathrm{dx}} \log \left(\log x^5\right) \\ & =\frac{1}{\log \left(\log x^5\right)} \times \frac{1}{\log \left(x^5\right)} \times \frac{\mathrm{d}}{\mathrm{dx}} \log x^5 \\ & =\frac{1}{\log \left(\log x^5\right)} \cdot \frac{1}{\log \left(x^5\right)} \cdot \frac{1}{x^5} \cdot \frac{\mathrm{~d}}{\mathrm{dx}} x^5 \\ & =\frac{1}{\log \left(\log x^5\right)} \cdot \frac{1}{\log \left(x^5\right)} \cdot \frac{1}{x^5} \cdot 5 x^4 \\ & =\frac{5}{x \log \left(x^5\right) \cdot \log \left(\log x^5\right)}\end{aligned}$

Hence, $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{5}{x \log \left(x^5\right) \cdot \log \left(\log x^5\right)}$

Question:29

Differentiate each of the following w.r.t. x
$\sin \sqrt{x}+\cos ^{2} \sqrt{x}$

Answer:

Let $\mathrm{y}=\sin \sqrt{x}+\cos ^2 \sqrt{x}$

Differentiating both sides w.r.t. x

$\begin{aligned} & \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}(\sin \sqrt{x})+\frac{\mathrm{d}}{\mathrm{dx}}\left(\cos ^2 \sqrt{x}\right) \\ & =\cos \sqrt{x} \cdot \frac{\mathrm{~d}}{\mathrm{dx}}(\sqrt{x})+2 \cos \sqrt{x} \cdot \frac{\mathrm{~d}}{\mathrm{dx}}(\cos \sqrt{x}) \\ & =\cos \sqrt{x} \cdot \frac{1}{2 \sqrt{x}}+2 \cos \sqrt{x}(-\sin \sqrt{x}) \cdot \frac{\mathrm{d}}{\mathrm{dx}} \sqrt{x} \\ & =\frac{1}{2 \sqrt{x}} \cdot \cos \sqrt{x}-2 \cos \sqrt{x} \cdot \sin \sqrt{x} \cdot \frac{1}{2 \sqrt{x}} \\ & =\frac{\cos \sqrt{x}}{2 \sqrt{x}}-\frac{\sin 2 \sqrt{x}}{2 \sqrt{x}}\end{aligned}$

Hence, $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\cos \sqrt{x}}{2 \sqrt{x}}-\frac{\sin 2 \sqrt{x}}{2 \sqrt{x}}$.

Question:30

Differentiate each of the following w.r.t. x
$sin^n (ax^2 + bx + c)$

Answer:

We have $sin^n (ax^2 + bx + c)$
$\begin{aligned} &y=\sin^n \left(a x^{2}+b x+c\right)\\ &\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\sin^n\left(\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}\right)\right)\\ &\text { since, we know, } x^{n}=n x^{n-1}\\ &\frac{d y}{d x}=n \cdot \sin^{n-1} \left(a x^{2}+b x+c\right) \cdot \frac{d}{d x} \sin \left(a x^{2}+b x+c\right)\\ &\frac{d y}{d x}=n \cdot \sin^{n-1} \left(a x^{2}+b x+c\right) \cdot \cos \left(a x^{2}+b x+c\right) \cdot \frac{d}{d x}\left(a x^{2}+b x+c\right)\\ &\frac{d y}{d x}=n \cdot \sin^{n-1} \left(a x^{2}+b x+c\right) \cdot \cos \left(a x^{2}+b x+c\right) \cdot(2 a x \cdot+b)\\ &\mathrm{y}^{\prime}=\mathrm{n}(2 \mathrm{ax} \cdot+\mathrm{b}) \cdot \sin ^{\mathrm{n}-1}\left(\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}\right) \cdot \cos \left(\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}\right) \end{aligned}$

Question:31

Differentiate each of the following w.r.t. x
$\cos (\tan \sqrt{x+1})$

Answer:

We have given $\cos (\tan \sqrt{x+1})$
Let us Assume $\sqrt{x+1}=w$
And $\tan \sqrt{x+1}=v$
$\mathrm{So}, \mathrm{y}=\cos \mathrm{v}$
Now, differentiate w.r.t v
$\frac{\mathrm{dy}}{\mathrm{d} \mathrm{v}}=(-\sin \mathrm{v})$
And, $\mathrm{v}=$ tan $\mathrm{w}$
Now, again differentiate w.r.t. w
$\frac{d v}{d w}=\sec ^{2} w$
And, we know, $\sqrt{x+1}=w$
So, differentiate w w.r.t. x we get
$\frac{d w}{d x}=\frac{1}{2 \sqrt{x+1}}$
Now, using the chain rule we get,
$\\\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{dy}}{\mathrm{dv}} \times \frac{\mathrm{dv}}{\mathrm{dw}} \times \frac{\mathrm{d} \mathrm{w}}{\mathrm{dx}}$ \\$\frac{d y}{d x}=(-\sin v) \times \sec ^{2} w \times \frac{1}{2 \sqrt{x+1}}$
Substitute the value of v and w
$\frac{\mathrm{dy}}{\mathrm{dx}}=\left(-\sin (\tan \sqrt{\mathrm{x}+1}) \times \sec ^{2} \sqrt{\mathrm{x}+1} \times \frac{1}{2 \sqrt{\mathrm{x}+1}}\right).$
Question:32 Differentiate each of the following w.r.t. x $sinx^2 + sin^2x + sin^2 (x^2)$

Answer:

Let us Assume $y=sinx^2 + sin^2x + sin^2 (x^2)$
$\begin{aligned} & \therefore \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}} \sin \left(x^2\right)+\frac{\mathrm{d}}{\mathrm{dx}}(\sin x)^2+\frac{\mathrm{d}}{\mathrm{dx}}\left(\sin x^2\right)^2 \\ & =\cos \left(x^2\right) \frac{\mathrm{d}}{\mathrm{dx}}\left(x^2\right)+2 \sin x \cdot \frac{\mathrm{~d}}{\mathrm{dx}}(\sin x)+2 \sin x^2 \frac{\mathrm{~d}}{\mathrm{dx}}\left(\sin x^2\right) \\ & =2 x \cos x^2+2 \sin x \cos x+2 \sin x^2 \cos x^2 \frac{\mathrm{~d}}{\mathrm{dx}} x^2 \\ & =2 x \cos x^2+2 \sin x \cos x+2 \sin x^2 \cos x^2 \times 2 x \\ & =2 \mathrm{x} \cos \left(\mathrm{x}^2\right)+\sin 2 \mathrm{x}+2 \mathrm{x} \sin 2\left(\mathrm{x}^2\right)\end{aligned}$

Question:33 Differentiate each of the following w.r.t. x
$\sin ^{-1}\left(\frac{1}{\sqrt{x+1}}\right)$

Answer:

Let $\mathrm{y}=\sin ^{-1} \frac{1}{\sqrt{x+1}}$

$\begin{aligned} & \therefore \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\sin ^{-1} \frac{1}{\sqrt{x+1}}\right) \\ & =\frac{1}{\sqrt{1-\left(\frac{1}{\sqrt{x+1}}\right)^2}} \cdot \frac{\mathrm{~d}}{\mathrm{dx}} \frac{1}{(x+1)^2} \\ & =\frac{1}{\sqrt{\frac{x+1-1}{x+1}}} \cdot \frac{\mathrm{~d}}{\mathrm{dx}}(x+1)^2 \\ & =\sqrt{\frac{x+1}{x}} \cdot \frac{-1}{2}(x+1)^{\frac{-3}{2}} \\ & =\frac{-1}{2 \sqrt{x}} \cdot\left(\frac{1}{x+1}\right)\end{aligned}$

Question:34 Differentiate each of the following w.r.t. x
$(sin x)^{cos x}$

Answer:

Given: $(sin x)^{cos x}$
To Find: Differentiate w.r.t x
We have $(sin x)^{cos x}$
Let $y=(sin x)^{cos x}$
Now, Taking Log on both sides, we get
Log y = cos x.log(sin x)
Now, Differentiate both sides w.r.t. x

$\begin{aligned} &\frac{1}{y} \frac{d y}{d x}=\cos x \cdot \frac{d}{d x}(\log (\sin x))+\log (\sin x) \cdot \frac{d}{d x}(\cos x)\\ &\text { By using product rule of differentiation }\\ &\frac{1}{y} \frac{d y}{d x}=\cos x \cdot \frac{1}{\sin x} \cdot \frac{d}{d x}(\sin x)+\log (\sin x) \cdot(-\sin x)\\ &\frac{1}{y} \frac{d y}{d x}=\cos x \cdot \frac{1}{\sin x}(\cos x)+\log (\sin x) \cdot(-\sin x)\\ &\frac{1}{y} \frac{d y}{d x}=\cos x \cdot \cot x-\log (\sin x) \cdot(\sin x)\\ &\frac{d y}{d x}=y[\cos x \cdot \cot x-\log (\sin x) \cdot(\sin x)] \end{aligned}$
Substitute the value of y, we get
$\begin{array}{l} y^{\prime}=(\sin x)^{\cos x}[\cos x \cdot \cot x-\sin x(\log \sin x)] \\ \text { Hence, } y^{\prime}=(\sin x)^{\cos x}[\cos x \cdot \cot x-\sin x(\log \sin x)] \end{array}$

Question:35

Differentiate each of the following w.r.t. x
$sin^mx . cos^nx$

Answer:

It is given $sin^mx. cos^nx$
$y=sin^mx . cos^nx$
$\begin{aligned} & \therefore \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left[(\sin x)^{\mathrm{m}} \cdot(\cos x)^{\mathrm{n}}\right] \\ & =(\sin x)^{\mathrm{m}} \frac{\mathrm{d}}{\mathrm{dx}}(\cos x)^{\mathrm{n}}+(\cos x)^{\mathrm{n}} \frac{\mathrm{d}}{\mathrm{dx}}(\sin x)^{\mathrm{m}} \\ & =(\sin x)^{\mathrm{m}} \mathrm{n}(\cos x)^{\mathrm{n}-1} \frac{\mathrm{~d}}{\mathrm{dx}}(\cos x)+(\cos x)^{\mathrm{n}} \mathrm{m}(\sin x)^{\mathrm{m}-1} \frac{\mathrm{~d}}{\mathrm{dx}}(\sin x) \\ & =(\sin x)^{\mathrm{m}} \mathrm{n}(\cos x)^{\mathrm{n}-1}(-\sin x)+(\cos x)^{\mathrm{n}} \mathrm{m}(\sin x)^{\mathrm{m}-1} \cos x \\ & =\sin ^{\mathrm{m}} \mathrm{x} \cos ^{\mathrm{n}} \mathrm{x}[-\mathrm{n} \tan \mathrm{x}+\mathrm{m} \cot \mathrm{x}]\end{aligned}$

Question:36 Differentiate each of the following w.r.t. x $(x + 1)^2 (x + 2)^3 (x + 3)^4$

Answer:

Let $y=(x+1)^2(x+2)^3(x+3)^4$

$\begin{aligned} & \therefore \log y=\log \left[(x+1)^2 \cdot(x+2)^3(x+3)^4\right] \\ & =2 \log (x+1)+3 \log (x+2)+4 \log (x+3)\end{aligned}$

Differentiating w.r.t. x both sides, we get

$\begin{aligned} & \frac{1}{y} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{2}{x+1}+\frac{3}{x+2}+\frac{4}{x+3} \\ & \therefore \frac{\mathrm{dy}}{\mathrm{dx}}=y\left[\frac{2}{x+1}+\frac{3}{x+2}+\frac{4}{x+3}\right] \\ & =(x+1)^2 \cdot(x+2)^3 \cdot(x+3)^4\left[\frac{2}{(x+1)}+\frac{3}{(x+2)}+\frac{4}{(x+3)}\right] \\ & =(x+1)^2 \cdot(x+2)^3 \cdot(x+3)^4 \times\left[\frac{2(x+3)(x+3)+3(x+1)(x+3)+4(x+1)(x+2)}{(x+1)(x+2)(x+3)}\right] \\ & =(\mathrm{x}+1)(\mathrm{x}+2)^2(\mathrm{x}+3)^3\left[9 \mathrm{x}^2+34 \mathrm{x}+29\right]\end{aligned}$

Question:37

Differentiate each of the following w.r.t. x
$\cos ^{-1}\left(\frac{\sin \mathrm{x}+\cos \mathrm{x}}{\sqrt{2}}\right), \frac{-\pi}{4}<\mathrm{x}<\frac{\pi}{4}$

Answer:

$\begin{aligned} &\cos ^{-1}\left(\frac{\sin x+\cos x}{\sqrt{2}}\right)\\ &=\cos ^{-1}\left(\sin \mathrm{x} \frac{1}{\sqrt{2}}+\cos \mathrm{x} \frac{1}{\sqrt{2}}\right)\\ &\text { As we know } \sin \pi / 4=\cos \pi / 4=\frac{1}{\sqrt{2}} \\ &=\cos ^{-1}\left(\sin x \sin \frac{\pi}{4}+\cos x \cos \frac{\pi}{4}\right)\\ &\text { We know } \cos (a-b)=\sin a \sin b+\cos a \cos b\\ &=\cos ^{-1}\left(\cos \left(\frac{\pi}{4}-\mathrm{x}\right)\right)\\ &=\left(\frac{\pi}{4}-\mathrm{x}\right) \end{aligned}$
$\begin{aligned} &\text { Now, }\\ &\frac{d y}{d x}=\frac{d}{d x}\left(\frac{\pi}{4}-x\right)\\ &=-1 \end{aligned}$

Question:38

Differentiate each of the following w.r.t. x
$\tan ^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right),-\frac{\pi}{4}<x<\frac{\pi}{4}$

Answer:

We have $\tan ^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right),-\frac{\pi}{4}<x<\frac{\pi}{4}$
$\begin{aligned} &\mathrm{y}=\tan ^{-1} \sqrt{\left(\frac{1-\cos \mathrm{x}}{1+\cos \mathrm{x}}\right)}\\ &\text { We know, }\\ &\cos x=\cos ^{2} \frac{x}{2}-\sin ^{2} \frac{x}{2}\\ &\mathrm{y}=\tan ^{-1} \sqrt{\left(\frac{\sin ^{2} \frac{\mathrm{x}}{2}+\cos ^{2} \frac{\mathrm{x}}{2}-\cos ^{2} \frac{\mathrm{x}}{2}+\sin ^{2} \frac{\mathrm{x}}{2}}{\sin ^{2} \frac{\mathrm{x}}{2}+\cos ^{2} \frac{\mathrm{x}}{2}+\cos ^{2} \frac{\mathrm{x}}{2}-\sin ^{2} \frac{\mathrm{x}}{2}}\right)}\\ &\mathrm{y}=\tan ^{-1} \sqrt{\left(\frac{2 \sin ^{2} \frac{\mathrm{x}}{2}}{2 \cos ^{2} \frac{\mathrm{x}}{2}}\right)}\\ &\mathrm{y}=\tan ^{-1} \sqrt{\left(\tan ^{2} \frac{\mathrm{x}}{2}\right)}\\ &y=\tan ^{-1}\left(\tan \frac{x}{2}\right) \end{aligned}$
$\begin{aligned} &\text { As the interval is }\\ &-\frac{\pi}{4}<x<\frac{\pi}{4}\\ &=\left\{\begin{array}{r} \tan ^{-1}\left(\tan \frac{x}{2}\right),-\frac{\pi}{4}<x<0 \\ \tan ^{-1}\left(\tan \frac{x}{2}\right), 0 \leq x<\frac{\pi}{4} \end{array}\right.\\ &=\left\{\begin{array}{c} -\frac{\mathrm{x}}{2},-\frac{\mathrm{\pi}}{4}<\mathrm{x}<0 \\ \frac{\mathrm{x}}{2}, 0 \leq \mathrm{x}<\frac{\pi}{4} \end{array}\right. \end{aligned}$
Differentiate w.r.t. x

$\frac{d y}{d x}=\left\{\begin{array}{c} -\frac{1}{2},-\frac{\pi}{4}<x<0 \\ \frac{1}{2}, 0 \leq x<\frac{\pi}{4} \end{array}\right.$

Question:39

Differentiate each of the following w.r.t. x
$\tan ^{-1}(\sec x+\tan x),-\frac{\pi}{4}<x<\frac{\pi}{2}$

Answer:

Let $\mathrm{y}=\tan ^{-1}(\sec \mathrm{x}+\tan \mathrm{x})$

Differentiating both sides w.r.t. x

$\begin{aligned} & \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left[\tan ^{-1}(\sec x+\tan x)\right] \\ & =\frac{1}{1+(\sec x+\tan x)^2} \cdot \frac{\mathrm{~d}}{\mathrm{dx}}(\sec x+\tan x) \\ & =\frac{1}{1+\sec ^2+\tan ^2 x+2 \sec x \tan x} \cdot\left(\sec x \tan x+\sec ^2 x\right) \\ & =\frac{1}{\left(1+\tan ^2 x\right)+\sec ^2 x+2 \sec x \tan x} \cdot \sec x(\tan x+\sec x) \\ & =\frac{1}{\sec ^2 x+\sec ^2 x+2 \sec x \tan x} \cdot \sec x(\tan x+\sec x) \\ & =\frac{1}{2 \sec ^2 x+2 \sec x \tan x} \cdot \sec x(\tan x+\sec x) \\ & =\frac{1}{2 \sec x(\sec x+\tan x)} \cdot \sec x(\tan x+\sec x) \\ & =\frac{1}{2}\end{aligned}$

Hence, $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2}$

Question:40

Differentiate each of the following w.r.t. x
$\tan ^{-1}\left(\frac{a \cos x-b \sin x}{b \cos x+a \sin x}\right),-\frac{\pi}{2}<x<\frac{\pi}{2} \text { and } \frac{a}{b} \tan x>-1$

Answer:

Let $\mathrm{y}=\tan ^{-1}\left(\frac{\mathrm{a} \cos x-\mathrm{b} \sin x}{\mathrm{~b} \cos x-\mathrm{a} \sin x}\right)$

$\begin{aligned} & \Rightarrow \mathrm{y}=\tan ^{-1}\left[\frac{\frac{\mathrm{a} \cos x}{\mathrm{~b} \cos x}-\frac{\mathrm{b} \sin x}{\mathrm{~b} \cos x}}{\frac{\mathrm{~b} \cos x}{\mathrm{~b} \cos x}+\frac{\mathrm{a} \sin x}{\mathrm{~b} \cos x}}\right] \\ & \Rightarrow \mathrm{y}=\tan ^{-1}\left[\frac{\frac{\mathrm{a}}{\mathrm{b}}-\tan x}{1+\frac{\mathrm{a}}{\mathrm{b}} \tan x}\right] \\ & \Rightarrow \mathrm{y}=\tan ^{-1} \frac{\mathrm{a}}{\mathrm{b}}-\tan ^{-1}(\tan x) \\ & \Rightarrow \mathrm{y}=\tan ^{-1} \frac{\mathrm{a}}{\mathrm{b}}-x\end{aligned}$

Differentiating both sides concerning x

$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\tan ^{-1} \frac{\mathrm{a}}{\mathrm{b}}\right)-\frac{\mathrm{d}}{\mathrm{dx}}(x)=0-1=-1$

Hence, $\frac{\mathrm{dy}}{\mathrm{dx}}=-1$.

Question:41

Differentiate each of the following w.r.t. x
$\sec ^{-1}\left(\frac{1}{4 x^{3}-3 x}\right), 0<x<\frac{1}{\sqrt{2}}$

Answer:

Let $\mathrm{y}=\sec ^{-1}\left(\frac{1}{4 x^3-3 x}\right)$

Put $x=\cos \theta$

$\begin{aligned} & \therefore \theta=\cos ^{-1} x \\ & y=\sec ^{-1}\left(\frac{1}{4 \cos ^3 \theta-3 \cos \theta}\right) \\ & \Rightarrow y=\sec ^{-1}\left(\frac{1}{\cos 3 \theta}\right) \ldots \ldots\left[\because \cos 3 \theta=4 \cos ^3 \theta-3 \cos \theta\right] \\ & \Rightarrow y=\sec ^{-1}(\sec 3 \theta) \\ & \Rightarrow y=3 \theta \\ & y=3 \cos ^{-1} x\end{aligned}$

Differentiating both sides w.r.t. X

$\begin{aligned} & \frac{d y}{d}=3 \cdot \frac{d}{d x} \cos ^{-1} x \\ & =3\left(\frac{-1}{\sqrt{1-x^2}}\right) \\ & =\frac{-3}{\sqrt{1-x^2}}\end{aligned}$

Hence, $\frac{d y}{d x}=\frac{-3}{\sqrt{1-x^2}}$.

Question:42 Differentiate each of the following w.r.t. x
$\tan ^{-1} \frac{3 \mathrm{a}^{2} \mathrm{x}-\mathrm{x}^{3}}{\mathrm{a}^{3}-3 \mathrm{ax}^{2}}, \frac{-1}{\sqrt{3}}<\frac{\mathrm{x}}{\sqrt{3}}<\frac{\mathrm{x}}{\mathrm{a}}<\frac{1}{\sqrt{3}}$

Answer:

$\\y=\tan ^{-1}\left(\frac{3 a^{2} x-x^{3}}{a^{3}-3 a x^{2}}\right)\\=\tan ^{-1}\left(\frac{3 \frac{x}{a}-\left(\frac{x}{a}\right)^{3}}{1-3\left(\frac{x}{a}\right)^{2}}\right)$
Let $x=a \tan \theta \Rightarrow \theta=\tan ^{-1} \frac{x}{a}$
$\\y=\tan ^{-1}\left[\frac{3 \tan \theta-\tan ^{3} \theta}{1-3 \tan ^{2} \theta}\right]\\=\tan ^{-1}(\tan 3 \theta)\\ =3 \theta\\=3 \tan ^{-1} \frac{x}{a} \\ \frac{d y}{d x}=3 \frac{d}{d x} \tan ^{-1} \frac{x}{a} \\ \quad=3\left[\frac{1}{1+\frac{x^{2}}{a^{2}}}\right] \cdot \frac{d}{d x}\left(\frac{x}{a}\right)\\=\frac{3 a^{2}}{a^{2}+x^{2}} \cdot \frac{1}{a}\\=\frac{3 a}{a^{2}+x^{2}}$

Question:43 Differentiate each of the following w.r.t. X $\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}\right), 1<x<1, x \neq 0$

Answer:

$\begin{aligned} &\text { We have given }\\ &\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}\right)\\ &y=\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}\right)\\ &\text { Put } x^{2}=\cos 2 \theta\\ &\mathrm{So} \end{aligned}$
$\\ \mathrm{y}=\tan ^{-1}\left(\frac{\sqrt{1+\cos 2 \theta}+\sqrt{1-\cos 2 \theta}}{\sqrt{1+\cos 2 \theta-\sqrt{1-\cos 2 \theta}}}\right) \\ \mathrm{y}=\tan ^{-1}\left(\frac{\sqrt{2 \cos ^{2} \theta}+\sqrt{2 \sin ^{2} \theta}}{\sqrt{2 \cos ^{2} \theta}-\sqrt{2 \sin ^{2} \theta}}\right) \\ \mathrm{y}=\tan ^{-1}\left(\frac{\sqrt{2} \cos \theta+\sqrt{2} \sin \theta}{\sqrt{2} \cos \theta-\sqrt{2} \sin \theta}\right) \\ \mathrm{y}=\tan ^{-1}\left(\frac{\cos \theta+\sin \theta}{\cos \theta-\sin \theta}\right)$
$\\ \mathrm{y}=\tan ^{-1}\left(\frac{\frac{\cos \theta}{\cos \theta}+\frac{\sin \theta}{\cos \theta}}{\frac{\cos \theta}{\cos \theta}-\frac{\sin \theta}{\cos \theta}}\right) \\ \mathrm{y}=\tan ^{-1}\left(\frac{1+\tan \theta}{1-\tan \theta}\right) \\ \mathrm{y}=\tan ^{-1}\left(\frac{\tan \frac{\pi}{4}+\tan \theta}{1-\tan \frac{\pi}{4} \tan \theta}\right) \\ \mathrm{y}=\tan ^{-1} \tan \left(\frac{\pi}{4}+\theta\right)$
$\begin{aligned} &\mathrm{y}=\frac{\pi}{4}+\theta\\ &\text { Differentiate, y w.r.t x }\\ &\frac{d y}{d x}=\frac{\pi}{4}+\frac{1}{2} \frac{d}{d x} \cos ^{-1} x^{2}\\ &\frac{d y}{d x}=0+\frac{1}{2} \cdot \frac{-1}{\sqrt{1-x^{4}}} \frac{d}{d x}\left(x^{2}\right)\\ &\frac{d y}{d x}=\frac{1}{2} \cdot \frac{-2 x}{\sqrt{1-x^{4}}}\\ &\text { Hence, } \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-\mathrm{x}}{\sqrt{1-\mathrm{x}^{4}}} \end{aligned}$

Question:44 Find $\frac{dy}{dx}$ of each of the functions expressed in parametric form in $x=t+\frac{1}{t}\: \: ,\: \: y=t-\frac{1}{t}$

Answer:

We have given two parametric equations,
$x=t+\frac{1}{t}\: \: ,\: \: y=t-\frac{1}{t}$
Now, differentiate both equations w.r.t x
We know, $\frac{d}{d x}\left(\frac{1}{x}\right)=-\frac{1}{x^{2}}$ and $\frac{d}{d x}(x)=1$
So,
$\frac{d x}{d t}=1-\frac{1}{t^{2}}$
and,
$\frac{d y}{d x}=1+\frac{1}{t^{2}}$
Now,
$\begin{aligned} &\frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}}\\ &\frac{d y}{d x}=\frac{1+\frac{1}{t^{2}}}{1-\frac{1}{t^{2}}}\\ &\text { Hence, }\\ &\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{t}^{2}+1}{\mathrm{t}^{2}-1} \end{aligned}$

Question:45

Find $\frac{dy}{dx}$ of each of the functions expressed in parametric form
$\mathrm{x}=\mathrm{e}^{\theta}\left(\theta+\frac{1}{\theta}\right), \mathrm{y}=\mathrm{e}^{-\theta}\left(\theta-\frac{1}{\theta}\right)$

Answer:

Given that, $\mathrm{x}=\mathrm{e}^\theta\left(\theta+\frac{1}{\theta}\right), \mathrm{y}=\mathrm{e}^{-\theta}\left(\theta-\frac{1}{\theta}\right)$

Differentiating both the parametric functions w.r.t. $\theta$

$\begin{aligned} & \frac{\mathrm{dx}}{\mathrm{d} \theta}=\mathrm{e}^\theta\left(1-\frac{1}{\theta^2}\right)+\left(\theta+\frac{1}{\theta}\right) \cdot \mathrm{e}^\theta \\ & \frac{\mathrm{dx}}{\mathrm{d} \theta}=\mathrm{e}^\theta\left(1-\frac{1}{\theta^2}+\theta+\frac{1}{\theta}\right) \\ & \Rightarrow \mathrm{e}^\theta\left(\frac{\theta^2-1+\theta^3+\theta}{\theta^2}\right) \\ & =\frac{\mathrm{e}^\theta\left(\theta^3+\theta^2+\theta-1\right)}{\theta^2} \\ & \mathrm{y}=\mathrm{e}^{-\theta}\left(\theta-\frac{1}{\theta}\right) \\ & \frac{\mathrm{dy}}{\mathrm{d} \theta}=\mathrm{e}^{-\theta}\left(1+\frac{1}{\theta^2}\right)+\left(\theta-\frac{1}{\theta}\right) \cdot\left(-\mathrm{e}^{-\theta}\right) \\ & \frac{\mathrm{dy}}{\mathrm{d} \theta}=\mathrm{e}^{-\theta}\left(1+\frac{1}{\theta^2}-\theta+\frac{1}{\theta}\right)\end{aligned}$

$\begin{aligned} & \Rightarrow \mathrm{e}^{-\theta}\left(\frac{\theta^2+1-\theta^3+\theta}{\theta^2}\right) \\ & =\mathrm{e}^{-\theta} \frac{\left(-\theta^3+\theta^2+\theta+1\right)}{\theta^2} \\ & \therefore \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{\mathrm{dy}}{\mathrm{d} \theta}}{\frac{\mathrm{d} x}{\mathrm{~d} \theta}} \\ & =\frac{\mathrm{e}^{-\theta}\left(\frac{-\theta^3+\theta^2+\theta+1}{\theta^2}\right)}{\mathrm{e}^\theta\left(\frac{\theta^3+\theta^2+\theta+1}{\theta^2}\right)} \\ & =\mathrm{e}^{-2 \theta}\left(\frac{-\theta^3+\theta^2+\theta+1}{\theta^3+\theta^2+\theta-1}\right)\end{aligned}$

Hence, $\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{e}^{-2 \theta}\left(\frac{-\theta^3+\theta^2+\theta+1}{\theta^3+\theta^2+\theta-1}\right)$

Question:46 Find dy/dx of each of the functions expressed in the parametric form in
$x=3 \cos \theta-2 \cos ^{3} \theta, y=3 \sin \theta-2 \sin ^{3} \theta$
Answer:

Given that: $\mathrm{x}=3 \cos \theta-2 \cos ^3 \theta$ and $\mathrm{y}=3 \sin \theta-2 \sin ^3 \theta$.

Differentiating both the parametric functions w.r.t. $\theta$

$\begin{aligned} & \frac{\mathrm{dx}}{\mathrm{d} \theta}=-3 \sin \theta-6 \cos ^2 \theta \cdot \frac{\mathrm{~d}}{\mathrm{~d} \theta}(\cos \theta) \\ & =-3 \sin \theta-6 \cos ^2 \theta \cdot(-\sin \theta) \\ & =-3 \sin \theta+6 \cos ^2 \theta \cdot \sin \theta \\ & \frac{\mathrm{dy}}{\mathrm{d} \theta}=3 o s \theta-6 \sin ^2 \theta \cdot \frac{\mathrm{~d}}{\mathrm{~d} \theta}(\sin \theta) \\ & ==3 \cos \theta-6 \sin ^2 \theta \cdot \cos \theta \\ & \therefore \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{d y}{\mathrm{~d} \theta}}{\frac{\mathrm{dx}}{\mathrm{d} \theta}}\end{aligned}$

$\begin{aligned} & =\frac{3 \cos \theta-6 \sin ^2 \theta \cos \theta}{-3 \sin \theta+6 \cos ^2 \theta \cdot \sin \theta} \\ & \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\cos \theta\left(3-6 \sin ^2 \theta\right)}{\sin \theta\left(-3+6 \cos ^2 \theta\right)} \\ & =\frac{\cos \theta\left[3-6\left(1-\cos ^2 \theta\right)\right]}{\sin \theta\left[-3+6 \cos ^2 \theta\right]} \\ & =\cot \theta\left(\frac{3-6+6 \cos ^2 \theta}{-3+6 \cos ^2 \theta}\right) \\ & =\cot \theta\left(\frac{-3+6 \cos ^2 \theta}{-3+6 \cos ^2 \theta}\right) \\ & =\cot \theta\end{aligned}$

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}}=\cot \theta$

Question:47

Find the dy/dx of each of the functions expressed in the parametric form in
$\sin \mathrm{x}=\frac{2 \mathrm{t}}{1+\mathrm{t}^{2}}, \tan \mathrm{y}=\frac{2 \mathrm{t}}{1-\mathrm{t}^{2}}$

Answer:

Given that $\sin \mathrm{x}=\frac{2 \mathrm{t}}{1+\mathrm{t}^2}$ and $\tan \mathrm{y}=\frac{2 \mathrm{t}}{1-\mathrm{t}^2}$

$\therefore$ Taking $\sin \mathrm{x}=\frac{2 \mathrm{t}}{1+\mathrm{t}^2}$

Differentiating both sides w.r.t t, we get

$\begin{aligned} & \cos x \cdot \frac{\mathrm{dx}}{\mathrm{dt}}=\frac{\left(1+\mathrm{t}^2\right) \cdot \frac{\mathrm{d}}{\mathrm{dt}}(2 \mathrm{t})-2 \mathrm{t} \cdot \frac{\mathrm{d}}{\mathrm{dt}}\left(1+\mathrm{t}^2\right)}{\left(1+\mathrm{t}^2\right)^2} \\ & \Rightarrow \cos x \cdot \frac{\mathrm{dx}}{\mathrm{dt}}=\frac{2\left(1+\mathrm{t}^2\right)-2 \mathrm{t} \cdot 2 \mathrm{t}}{\left(1+\mathrm{t}^2\right)^2} \\ & \Rightarrow \frac{\mathrm{dx}}{\mathrm{dt}}=\frac{2+2 \mathrm{t}^2-4 \mathrm{t}^2}{\left(1-\mathrm{t}^2\right)^2} \times \frac{1}{\cos x} \\ & \Rightarrow \frac{\mathrm{dx}}{\mathrm{dt}}=\frac{2-2 \mathrm{t}^2}{\left(1+\mathrm{t}^2\right)^2} \times \frac{1}{\sqrt{1-\sin ^2 x}} \\ & \Rightarrow \frac{\mathrm{dx}}{\mathrm{dt}}=\frac{2\left(1-\mathrm{t}^2\right)}{\left(1+\mathrm{t}^2\right)^2} \times \frac{1}{\sqrt{1-\left(\frac{2 \mathrm{t}}{1+\mathrm{t}^2}\right)^2}}\end{aligned}$

$\begin{aligned} & \Rightarrow \frac{d x}{d t}=\frac{2\left(1-t^2\right)}{\left(1+t^2\right)^2} \times \frac{1}{\frac{\sqrt{\left(1+t^2\right)^2-4 t^2}}{\left(1+t^2\right)^2}} \\ & \Rightarrow \frac{d x}{d t}=\frac{2\left(1-t^2\right)}{\left(1+t^2\right)^2} \times \frac{1+t^2}{\sqrt{1+t^4+2 t^2-4 t^2}} \\ & \Rightarrow \frac{d x}{d t}=\frac{2\left(1-t^2\right)}{\left(1+t^2\right)^2} \times \frac{1}{\sqrt{1+t^4-2 t^2}} \\ & \Rightarrow \frac{d x}{d t}=\frac{2\left(1-t^2\right)}{\left(1+t^2\right)^2} \times \frac{1}{\sqrt{\left(1-t^2\right)^2}}\end{aligned}$

$\begin{aligned} & \Rightarrow \frac{\mathrm{dx}}{\mathrm{dt}}=\frac{2\left(1-\mathrm{t}^2\right)}{\left(1+\mathrm{t}^2\right)^2} \times \frac{1}{\left(1-\mathrm{t}^2\right)} \\ & \Rightarrow \frac{\mathrm{dx}}{\mathrm{dt}}=\frac{2}{1+\mathrm{t}^2}\end{aligned}$

Now taking, $\tan \mathrm{y}=\frac{2}{1-\mathrm{t}^2}$

Differentiating both sides w.r.t, t , we get

$\begin{aligned} & \frac{\mathrm{d}}{\mathrm{dt}}(\tan y)=\frac{\mathrm{d}}{\mathrm{dt}}\left(\frac{2 \mathrm{t}}{1-\mathrm{t}^2}\right) \\ & \Rightarrow \sec ^2 y \frac{\mathrm{dy}}{\mathrm{dt}}=\frac{\left(1-\mathrm{t}^2\right) \cdot \frac{\mathrm{d}}{\mathrm{dt}}(2 \mathrm{t})-2 \mathrm{t} \cdot \frac{\mathrm{d}}{\mathrm{dt}}\left(1-\mathrm{t}^2\right)}{\left(1-\mathrm{t}^2\right)^2} \\ & \Rightarrow \sec ^2 y \frac{\mathrm{dy}}{\mathrm{dt}}=\frac{\left(1-\mathrm{t}^2\right) \cdot 2-2 \mathrm{t} \cdot(-2 \mathrm{t})}{\left(1-\mathrm{t}^2\right)^2} \\ & \Rightarrow \sec ^2 y \frac{\mathrm{dy}}{\mathrm{dt}}=\frac{2-2 \mathrm{t}^2+4 \mathrm{t}^2}{\left(1-\mathrm{t}^2\right)^2} \\ & \Rightarrow \frac{\mathrm{dy}}{\mathrm{dt}}=\frac{2+2 \mathrm{t}^2}{\left(1-\mathrm{t}^2\right)^2} \times \frac{1}{\sec ^2 y}\end{aligned}$

$\begin{aligned} & \Rightarrow \frac{d y}{d t}=\frac{2\left(1+t^2\right)}{\left(1-t^2\right)^2} \times \frac{1}{1+\tan ^2 y} \\ & \Rightarrow \frac{d y}{d t}=\frac{2\left(1+\mathrm{t}^2\right)}{\left(1-\mathrm{t}^2\right)^2} \times \frac{1}{1+\left(\frac{2 \mathrm{t}}{1-\mathrm{t}^2}\right)^2} \\ & \Rightarrow \frac{\mathrm{dy}}{\mathrm{dt}}=\frac{2\left(1+\mathrm{t}^2\right)}{\left(1-\mathrm{t}^2\right)^2} \times \frac{1}{\frac{\left(1-\mathrm{t}^2\right)^2+4 \mathrm{t}^2}{\left(1-\mathrm{t}^2\right)^2}} \\ & \Rightarrow \frac{\mathrm{dy}}{\mathrm{dt}}=\frac{2\left(1+\mathrm{t}^2\right)}{\left(1-\mathrm{t}^2\right)^2} \times \frac{\left(1-\mathrm{t}^2\right)^2}{1+\mathrm{t}^2+2 \mathrm{t}^2+4 \mathrm{t}^2} \\ & \Rightarrow \frac{\mathrm{dy}}{\mathrm{dt}}=\frac{2\left(1+\mathrm{t}^2\right)}{\left(1-\mathrm{t}^2\right)^2} \times \frac{\left(1-\mathrm{t}^2\right)^2}{1+\mathrm{t}^4+2 \mathrm{t}^2}\end{aligned}$

$\begin{aligned} & \Rightarrow \frac{d y}{d t}=\frac{2\left(1+t^2\right)}{\left(1-t^2\right)^2} \times \frac{\left(1-t^2\right)^2}{\left(1+t^2\right)^2} \\ & \Rightarrow \frac{d y}{d t}=\frac{2}{1+t^2} \\ & \therefore \frac{d y}{d t}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}} \\ & =\frac{\frac{2}{1+t^2}}{\frac{2}{1+t^2}} \\ & =1\end{aligned}$

Hence $\frac{\mathrm{dy}}{\mathrm{dt}}=1$

Question:48 Find dy/dx of each of the functions expressed in parametric form
$\mathrm{x}=\frac{1+\log \mathrm{t}}{\mathrm{t}^{2}}, \mathrm{y}=\frac{3+2 \log \mathrm{t}}{\mathrm{t}}$

Answer:

Given that: $\mathrm{x}=\frac{1+\log \mathrm{t}}{\mathrm{t}^2}, \mathrm{y}=\frac{3+2 \log \mathrm{t}}{\mathrm{t}}$

Differentiating both the parametric functions w.r.t. t

$\begin{aligned} & \frac{d x}{d t}=\frac{t^2 \cdot \frac{d}{d t}(1+\log t)-(1+\log t) \cdot \frac{d}{d t}\left(t^2\right)}{t^4} \\ & =\frac{t^2 \cdot\left(\frac{1}{t}\right)-(1+\log t) \cdot 2 t}{t^4} \\ & =\frac{t-(1+\log t) \cdot 2 t}{t^4} \\ & =\frac{t[1-2-2 \log t]}{t^4} \\ & =\frac{-(1+2 \log t)}{t^3} \\ & y=\frac{3+2 \log t}{t}\end{aligned}$

$\begin{aligned} & \frac{d y}{d t}=\frac{t \cdot \frac{d}{d t}(3+2 \log t)-(3+2 \log t) \cdot \frac{d}{d t}(t)}{t^2} \\ & =\frac{t\left(\frac{2}{t}\right)-(3+2 \log t) \cdot 1}{t^2} \\ & =\frac{2-3-2 \log t}{t^2} \\ & =\frac{-(1+2 \log t)}{t^2} \\ & \therefore \frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}} \\ & =\frac{-(1+2 \log t)}{t^2} \\ & \frac{-(1+2 \log t)}{t^3} \\ & =\frac{t^3}{t^2} \\ & =t\end{aligned}$

Hence, $\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{t}$.

Question:49

If $\mathrm{x}=e^{\cos 2 t}$ and $\mathrm{y}=\mathrm{e}^{ \sin 2 t}$, prove that $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-\mathrm{y} \log \mathrm{x}}{\mathrm{x} \log \mathrm{y}} \text { . }$

Answer:

Given that: $e^{\cos 2 t}$ and $y=e^{\sin 2 t}$

$\Rightarrow \cos 2 t=\log x$ and $\sin 2 t=\log y$

Differentiating both the parametric functions w.r.t. T

$\begin{aligned} & \frac{d x}{d t}=e^{\cos 2 t} \cdot \frac{d}{d t}(\cos 2 t) \\ & =e^{\cos 2 t}(-\sin 2 t) \cdot \frac{d}{d t}(2 t) \\ & =-e^{\cos 2 t} \cdot \sin 2 t \cdot 2 \\ & =2 e^{\cos 2 t} \cdot \sin 2 t\end{aligned}$

Now $y=e^{\sin 2 t}$

$\begin{aligned} & \frac{d y}{d t}=e^{\sin 2 t} \cdot \frac{d}{d t}(\sin 2 t) \\ & =e^{\sin 2 t} \cdot \cos 2 t \cdot \frac{d}{d t}(2 t) \\ & =e^{\sin 2 t} \cdot \cos 2 t \cdot 2 \\ & =2 e^{\sin 2 t} \cdot \cos 2 t \\ & \therefore \frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}} \\ & =\frac{2 e^{\sin 2 t} \cdot \cos 2 t}{-2 e^{\cos 2 t} \cdot \sin 2 t} \\ & =\frac{e^{\sin 2 t} \cdot \cos 2 t}{-e^{\cos 2 t} \cdot \sin 2 t} \\ & =\frac{y \cos 2 t}{-x \sin 2 t}\end{aligned}$

$=\frac{y \log x}{-x \log y}$

Hence, $\frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{y \log x}{x \log y}$.

Question:50

If $x = a\sin 2t (1 + \cos2t) $and y$ = b\cos2t(1 - \cos2t)$, show that $\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{\mathrm{at} \mathrm{t}=\frac{\pi}{4}}=\frac{\mathrm{b}}{\mathrm{a}}$

Answer:

$x = a\sin 2t (1 + \cos2t) $and y$ = b\cos2t(1 - \cos2t)$
Differentiate w.r.t t
$\\ x=a \sin 2 t(1+\cos 2 t) \\ \frac{d x}{d t}=\frac{d}{d t}[\operatorname{asin} 2 t(1+\cos 2 t)] \\ \frac{d x}{d t}=a \frac{d}{d t}[\sin 2 t(1+\cos 2 t)] \\ \frac{d x}{d t}=a\left[\sin 2 t \frac{d}{d t}(1+\cos 2 t)+(1+\cos 2 t) \frac{d}{d t}(\sin 2 t)\right] \\ \frac{d x}{d t}=a[\sin 2 t \cdot(-2 \sin 2 t)+(1+\cos 2 t) \cdot 2 \cos 2 t] \\ \frac{d x}{d t}=-2 a\left[\sin ^{2} 2 t-\cos 2 t(1+\cos 2 t)\right]$
Also,
y = bcos2t
$\\ \frac{\mathrm{dy}}{\mathrm{dt}}=\frac{\mathrm{d}}{\mathrm{dt}}(\mathrm{b} \cos 2 \mathrm{t}) \\ \frac{\mathrm{dy}}{\mathrm{dt}}=\mathrm{b} \frac{\mathrm{d}}{\mathrm{dt}}(\cos 2 \mathrm{t}) \\ \frac{\mathrm{dy}}{\mathrm{dt}}=\mathrm{b}\left[-\sin 2 \mathrm{t} \cdot \frac{\mathrm{d}}{\mathrm{dt}}(2 \mathrm{t})\right] \\ \frac{\mathrm{dy}}{\mathrm{dt}}=\mathrm{b}[-2 \sin 2 \mathrm{t}]$
Now, for dy/dx

$\\ \frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}} \\ \frac{d y}{d x}=\frac{b[-2 \sin 2 t]}{-2 a\left[\sin ^{2} 2 t-\cos 2 t(1+\cos 2 t)\right]} \\ \frac{d y}{d x}=\frac{b[\sin 2 t]}{a\left[\sin ^{2} 2 t-\cos 2 t(1+\cos 2 t)\right]} \\ \frac{d y}{d x} \text { at } t=\frac{\pi}{4}$
$\\ \frac{d y}{d x}=\frac{b\left[\sin \frac{2 \pi}{4}\right]}{a\left[\sin ^{2} \frac{2 \pi}{4}-\cos \frac{2 \pi}{4}\left(1+\cos \frac{2 \pi}{4}\right)\right]} \\ \frac{d y}{d x}=\frac{b\left[\sin \frac{\pi}{2}\right]}{a\left[\sin ^{2} \frac{\pi}{2}-\cos \frac{\pi}{2}\left(1+\cos \frac{\pi}{2}\right)\right]} \\ \frac{d y}{d x}=\frac{b[1]}{a[1-0(1+0)]} \\ \frac{d y}{d x}=\frac{b}{a}$
Hence Proved.

Question:51

If , find $\frac{\mathrm{dy}}{\mathrm{dx}} \text { at } \mathrm{t}=\frac{\pi}{3}$

Answer:

$x = 3\sin t - \sin 3t, y = 3\cos t - \cos 3t$
Differentiate w.r.t t in both equation
x = 3sint - sin3t

$\begin{aligned} &\frac{d x}{d t}=\frac{d}{d t}(3 \sin t-\sin 3 t)\\ &\frac{d x}{d t}=\frac{d}{d t}(3 \sin t)-\frac{d}{d t}(\sin 3 t)\\ &\left.\frac{d x}{d t}=3 \cos t-3 \cos 3 t\right.\\ &\text { Now, for y }\\ &y=3 \operatorname{cost}-\cos 3 t \end{aligned}$
$\\ \frac{\mathrm{dy}}{\mathrm{dt}}=\frac{\mathrm{d}}{\mathrm{dt}}(3 \cos \mathrm{t}- \cos 3 \mathrm{t}) \\ \frac{\mathrm{dy}}{\mathrm{dt}}=3(-\sin \mathrm{t})- \frac{\mathrm{d}}{\mathrm{dt}}(\cos 3 \mathrm{t}) \\ \frac{\mathrm{dy}}{\mathrm{dt}}=3(-\sin \mathrm{t})-3(-\sin 3 \mathrm{t}) \\ \frac{\mathrm{dy}}{\mathrm{dt}}=-3 \sin \mathrm{t}+3 \sin 3 \mathrm{t} \\ \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{\mathrm{dy}}{\mathrm{dt}}}{\frac{\mathrm{d} \mathrm{x}}{\mathrm{dt}}}$
$\\ \frac{d y}{d x}=\frac{-3 \sin t+3 \sin 3 t}{3 \cos t-3 \cos 3 t} \\ \text { At } t=\pi / 3 \\ \frac{d y}{d x}=\frac{-3 \sin \frac{\pi}{3}+3 \sin \frac{3 \pi}{3}}{3 \cos \frac{\pi}{3}-3 \cos \frac{3 \pi}{3}} \\ \frac{d y}{d x}=\frac{-\sin \frac{\pi}{3}+\sin \pi}{\cos \frac{\pi}{3}-\cos \pi} \\ \frac{d y}{d x}=\frac{-\frac{\sqrt{3}}{2}+0}{\frac{1}{2}-(-1)}$
$\\ \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}+1} \\ \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-\frac{\sqrt{3}}{2}}{\frac{3}{2}} \\ \frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{\sqrt{3}}{2} \times \frac{2}{3} \\ \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-1}{\sqrt{3}}$

Question:52

Differentiate $\frac{x}{\sin x}$ w.r.t. sinx.

Answer:

Let us Assume,
$\begin{aligned} &\mathrm{u}=\frac{\mathrm{x}}{\sin \mathrm{x}}, \mathrm{v}=\sin \mathrm{x}\\ &\text { Now, differentiate w.r.t } x\\ &\frac{d u}{d x}=\frac{\sin x \cdot \frac{d}{d x}(x)-x \cdot \frac{d}{d x}(\sin x)}{(\sin x)^{2}}\\ &\frac{d u}{d x}=\frac{\sin x-x \cdot \cos x}{(\sin x)^{2}}\\ &\text { And, } v=\sin x\\ &\frac{\mathrm{d} \mathrm{v}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}(\sin \mathrm{x}) \end{aligned}$
$\begin{aligned} &\frac{d v}{d x}=\cos x\\ &\text { Now, }\\ &\frac{\mathrm{du}}{\mathrm{dv}}=\frac{\frac{\mathrm{du}}{\mathrm{dx}}}{\frac{\mathrm{dv}}{\mathrm{dx}}}\\ &\frac{\mathrm{du}}{\mathrm{dv}}=\frac{\sin \mathrm{x} \cdot-\mathrm{x} \cdot \cos \mathrm{x}}{(\sin \mathrm{x})^{2}} \times \frac{1}{\cos \mathrm{x}}\\ &\frac{d u}{d v}=\frac{\tan x-x}{\sin ^{2} x} \end{aligned}$

Question:53

Differentiate $\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}-1}{x}\right)$ w.r.t. $tan^{-1}x$ when $x\neq 0$

Answer:

We have $\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}-1}{x}\right)$
Let us Assume,
$\begin{aligned} &\mathrm{p}=\tan ^{-1}\left(\frac{\sqrt{1+\mathrm{x}^{2}}-1}{\mathrm{x}}\right) \text { and } \theta=\tan ^{-1}{ \mathrm{x}}\\ &\text { And, put } x=\tan \theta\\ &p=\tan ^{-1} \frac{\sqrt{1+\tan ^{2} \theta}-1}{\tan \theta}\\ &p=\tan ^{-1} \frac{\sqrt{\sec ^{2} \theta}-1}{\tan \theta}\\ &p=\tan ^{-1} \frac{\sec \theta-1}{\tan \theta} \end{aligned}$
$\\ \mathrm{p}=\tan ^{-1}\left(\frac{1-\cos \theta}{\sin \theta}\right) \\ \mathrm{p}=\tan ^{-1}\left(\frac{2 \sin ^{2} \frac{\theta}{2}}{2 \sin \frac{\theta}{2} \cdot \cos \frac{\theta}{2}}\right) \\ \mathrm{p}=\tan ^{-1}\left(\frac{\sin \frac{\theta}{2}}{\cos \frac{\theta}{2}}\right) \\ \mathrm{p}=\tan ^{-1}\left(\tan \frac{\theta}{2}\right) \\ \mathrm{p}=\frac{\theta}{2}$
$\frac{dp}{d\theta} = \frac{1}{2}$
Hence Differentiation of $\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}-1}{x}\right)$w.r.t. $\tan^{-1}x$ is $\frac{1}{2}$.

Question:54

Find $\frac{dy}{dx}$ when x and y are connected by the relation given

$\sin (x y)+\frac{x}{y}=x^{2}-y$

Answer:

We have,
$\sin (x y)+\frac{x}{y}=x^{2}-y$
Use the chain rule and quotient rule to get:
Chain Rule
f(x) = g(h(x))
f’(x) = g’(h(x))h’(x)
By Quotient Rule
$\frac{\mathrm{d}}{\mathrm{d} \mathrm{x}}\left[\frac{\mathrm{f}(\mathrm{x})}{\mathrm{g}(\mathrm{x})}\right]=\frac{\mathrm{g}(\mathrm{x}) \mathrm{f}^{\prime}(\mathrm{x})-\mathrm{f}(\mathrm{x}) \mathrm{g}^{\prime}(\mathrm{x})}{[\mathrm{g}(\mathrm{x})]^{2}}$
On differentiating both the sides concerning x, we get

$\begin{aligned} &\cos x y \times \frac{d}{d x}(x y)+\frac{y \frac{d}{d x}(x)-x \frac{d}{d x}(y)}{y^{2}}=2 x-\frac{d y}{d x}\\ &\text { By product rule: }\\ &\frac{\mathrm{d}}{\mathrm{dx}}[\mathrm{f}(\mathrm{x}) \mathrm{g}(\mathrm{x})]=\mathrm{f}(\mathrm{x}) \mathrm{g}^{\prime}(\mathrm{x})+\mathrm{g}(\mathrm{x}) \mathrm{f}^{\prime}(\mathrm{x})\\ &\left[\because \frac{d}{d x} \sin x=\cos x\right] \end{aligned}$
$\begin{aligned} &\Rightarrow \cos (x y)\left[x \frac{d y}{d x}+y \times(1)\right]+\frac{y \times 1-x \frac{d y}{d x}}{y^{2}}=2 x-\frac{d y}{d x}\\ &\text { Multiplying by } y^{2} \text { to both the sides, we get }\\ &\Rightarrow \cos (x y)\left[x y^{2} \frac{d y}{d x}+y^{3}\right]+y-x \frac{d y}{d x}=2 x y^{2}-y^{2} \frac{d y}{d x}\\ &\Rightarrow x y^{2} \cos (x y) \frac{d y}{d x}+y^{3} \cos (x y)+y-x \frac{d y}{d x}=2 x y^{2}-y^{2} \frac{d y}{d x}\\ &\Rightarrow \mathrm{xy}^{2} \cos (\mathrm{xy}) \frac{\mathrm{dy}}{\mathrm{dx}}-\mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}^{2} \frac{\mathrm{dy}}{\mathrm{dx}}=2 \mathrm{xy}^{2}-\mathrm{y}^{3} \cos (\mathrm{xy})-\mathrm{y}\\ &\Rightarrow \frac{d y}{d x}\left[x y^{2} \cos (x y)-x+y^{2}\right]=2 x y^{2}-y^{3} \cos (x y)-y\\ &\Rightarrow \frac{d y}{d x}=\frac{2 x y^{2}-y^{3} \cos (x y)-y}{x y^{2} \cos (x y)-x+y^{2}} \end{aligned}$

Question:55

Find $\frac{dy}{dx}$ when x and y are connected by the relation given

$sec (x + y) = xy$

Answer:

We have,
sec(x + y) = xy
By the rules given below:
Chain Rule
f(x) = g(h(x))
f’(x) = g’(h(x))h’(x)
Product rule:
$\begin{aligned} &\frac{\mathrm{d}}{\mathrm{d} \mathrm{x}}[\mathrm{f}(\mathrm{x}) \mathrm{g}(\mathrm{x})]=\mathrm{f}(\mathrm{x}) \mathrm{g}^{\prime}(\mathrm{x})+\mathrm{g}(\mathrm{x}) \mathrm{f}^{\prime}(\mathrm{x})\\ &\text { On differentiating both sides with respect to } x, \text { we get }\\ &\sec (x+y) \tan (x+y) \frac{d}{d x}(x+y)=y+x \frac{d}{d x} y\\ &\left[\because \frac{d}{d x} \sec (x)=\sec x \tan x\right]\\ &\Rightarrow \sec (\mathrm{x}+\mathrm{y}) \tan (\mathrm{x}+\mathrm{y})\left[1+\frac{\mathrm{d} y}{\mathrm{~d} \mathrm{x}}\right]=\mathrm{y}+\mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}} \end{aligned}$
$\\ \Rightarrow \sec (x+y) \tan (x+y)+\sec (x+y) \tan (x+y) \frac{d y}{d x}=y+x \frac{d y}{d x} \\ \Rightarrow \sec (x+y) \tan (x+y) \frac{d y}{d x}-x \frac{d y}{d x}=y-\sec (x+y) \tan (x+y) \\ \Rightarrow \frac{d y}{d x}[\sec (x+y) \tan (x+y)-x]=y-\sec (x+y) \tan (x+y) \\ \therefore \frac{d y}{d x}=\frac{y-\sec (x+y) \tan (x+y)}{\sec (x+y) \tan (x+y)-x}$

Question:58

If $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$, then show that $\frac{d y}{d x} \cdot \frac{d x}{d y}=1$

Answer:

Given: $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$
Differentiating the above concerning x, we get
$\\ 2 \mathrm{ax}+2 \mathrm{~h}\left[\mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}\right]+\mathrm{b} \frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{y}^{2}\right)+2 \mathrm{~g}+2 \mathrm{f} \frac{\mathrm{d}}{\mathrm{dx}} \mathrm{y}=0 \\ \Rightarrow 2 \mathrm{ax}+2 \mathrm{hx} \frac{\mathrm{dy}}{\mathrm{dx}}+2 \mathrm{hy}+2 \mathrm{by} \frac{\mathrm{dy}}{\mathrm{dx}}+2 \mathrm{~g}+2 \mathrm{f} \frac{\mathrm{dy}}{\mathrm{dx}}=0 \\ \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}[2 \mathrm{hx}+2 \mathrm{by}+2 \mathrm{f}]=-2 \mathrm{ax}-2 \mathrm{hy}-2 \mathrm{~g} \\ \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-[2 \mathrm{ax}+2 \mathrm{hy}+2 \mathrm{~g}]}{2 \mathrm{hx}+2 \mathrm{by}+2 \mathrm{f}} \ldots(\mathrm{ii})$
Now, we again differentiate eq (i) concerning y, we get,

$\\ a \frac{d}{d y}\left(x^{2}\right)+2 h\left[x+y \frac{d}{d y} x\right]+2 y b+2 g \frac{d x}{d y}+2 f=0 \\ \Rightarrow 2 a x \frac{d x}{d y}+2 h x+2 h y \frac{d x}{d y}+2 b y+2 g \frac{d x}{d y}+2 f=0 \\ \Rightarrow \frac{d x}{d y}[2 a x+2 h y+2 g]=-2 h x-2 b y-2 f \\ \Rightarrow \frac{d x}{d y}=\frac{-[2 h x+2 b y+2 f]}{2 a x+2 h y+2 g} \ldots(i i i)$
Now, by multiplying Eq. (ii) and (iii), we get

$\\ \Rightarrow \frac{d y}{d x} \times \frac{d x}{d y}=\frac{-[2 a x+2 h y+2 g]}{2 h x+2 b y+2 f} \times \frac{-[2 h x+2 b y+2 f]}{2 a x+2 h y+2 g} =1$
Hence Proved

Question:59

If $x=e^{x/y}$prove that $\frac{dy}{dx}= \frac{x-y}{x\log x}$

Answer:

Given that: $\mathrm{x}=\mathrm{e}^{\frac{x}{y}}$

Taking $\log$ on both the sides,

$\begin{aligned} & \log \mathrm{x}=\log \mathrm{e}^{\frac{x}{y}} \\ & \Rightarrow \log \mathrm{x}=\frac{x}{y} \log \mathrm{e}\end{aligned}$

$\Rightarrow \log \mathrm{x}=\frac{x}{y}$

Differentiating both sides w.r.t. X

$\begin{aligned} & \frac{\mathrm{d}}{\mathrm{dx}} \log x=\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{x}{y}\right) \\ & \Rightarrow \frac{1}{x}=\frac{y \cdot 1-x \cdot \frac{\mathrm{dy}}{\mathrm{dx}}}{y^2} \\ & \Rightarrow \mathrm{y}^2=x y-x^2 \cdot \frac{\mathrm{dy}}{\mathrm{dx}} \\ & \Rightarrow x^2 \cdot \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{xy}-\mathrm{y}^2 \\ & \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{y(x-y)}{x^2} \\ & \Rightarrow \frac{\mathrm{dy}}{\mathrm{d}}=\frac{y}{x} \cdot\left(\frac{x-y}{x}\right) \\ & \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{\log x} \cdot\left(\frac{x-y}{x}\right)\end{aligned}$

Hence, ${ }^{\prime}$ "dy"/"dx" = (x - y)/(xlogx).

Question:60

If $y^x = e^{y-x}$, prove that $\frac{d y}{d x}=\frac{(1+\log y)^{2}}{\log y}$

Answer:

Given that: $\mathrm{y}^{\mathrm{x}}=\mathrm{e}^{\mathrm{y}-\mathrm{x}}$

Taking $\log$ on both sides $\log \mathrm{y}^{\mathrm{x}}=\log \mathrm{e}^{\mathrm{y}-\mathrm{x}}$

$\begin{aligned} & \Rightarrow \mathrm{x} \log \mathrm{y}=(\mathrm{y}-\mathrm{x}) \log \mathrm{e} \\ & \Rightarrow \mathrm{x} \log \mathrm{y}=\mathrm{y}-\mathrm{x} \quad \ldots . .[\because \log \mathrm{e}=1] \\ & \Rightarrow \mathrm{x} \log \mathrm{y}+\mathrm{x}=\mathrm{y} \\ & \Rightarrow \mathrm{x}(\log \mathrm{y}+1)=\mathrm{y} \\ & \Rightarrow \mathrm{x}=\frac{y}{\log y+1}\end{aligned}$

Differentiating both sides w.r.t. Y

$\begin{aligned} & \frac{\mathrm{dx}}{\mathrm{dy}}=\frac{\mathrm{d}}{\mathrm{dy}}\left(\frac{y}{\log y+1}\right) \\ & =\frac{(\log y+1) \cdot 1-y \cdot \frac{\mathrm{~d}}{\mathrm{dy}}(\log y+1)}{(\log y+1)^2} \\ & =\frac{\log y+1-y \cdot \frac{1}{2}}{(\log y+1)^2} \\ & =\frac{\log y}{(\log y+1)^2}\end{aligned}$

We know that

$\begin{aligned} & \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{\frac{\mathrm{dx}}{\mathrm{dy}}} \\ & =\frac{1}{\frac{\log y}{(\log y+1)^2}} \\ & =\frac{(\log y+1)^2}{\log y}\end{aligned}$

Hence, $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{(\log y+1)^2}{\log y}$.

Question:61

If $\mathrm{y}=(\cos \mathrm{x})^{(\cos \mathrm{x})^{(\cos \mathrm{x}) \ldots . \infty}}$Show that $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{y}^{2} \tan \mathrm{x}}{\mathrm{y} \log \cos \mathrm{x}-1}$

Answer:

Given that $\mathrm{y}=(\cos x)^{(\cos x)^{(\cos x) \ldots \infty}}$,

$\Rightarrow \mathrm{y}=(\cos \mathrm{x})^{\mathrm{y}} \ldots . .\left[y=(\cos x)^{\left.(\cos x)^{(\cos x) \ldots \infty}\right]}\right]$

Taking $\log$ on both sides $\log y=y \cdot \log (\cos x)$

Differentiating both sides w.r.t. x

$\begin{aligned} & \frac{1}{y} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}=y \cdot \frac{\mathrm{~d}}{\mathrm{dx}} \log (\cos x)+\log (\cos x) \cdot \frac{\mathrm{dy}}{\mathrm{dx}} \\ & \Rightarrow \frac{1}{y} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}=y \cdot \frac{1}{\cos x} \cdot \frac{\mathrm{~d}}{\mathrm{dx}}(\cos x)+\log (\cos x) \cdot \frac{\mathrm{dy}}{\mathrm{dx}} \\ & \Rightarrow \frac{1}{y} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}=y \cdot \frac{1}{\cos x} \cdot(-\sin x)+\log (\cos x) \cdot \frac{\mathrm{dy}}{\mathrm{dx}} \\ & \Rightarrow \frac{1}{y} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}-\log (\cos x) \frac{\mathrm{dy}}{\mathrm{dx}}=-\mathrm{y} \tan \mathrm{x} \\ & \Rightarrow\left[\frac{1}{y}-\log (\cos x)\right] \frac{\mathrm{dy}}{\mathrm{dx}}=-\mathrm{y} \tan \mathrm{x} \\ & \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-y \tan x}{\frac{1}{y}-\log (\cos x)} \\ & =\frac{y^2 \tan x}{y \log \cos x-1}\end{aligned}$

Hence, $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{y^2 \tan x}{y \log \cos x-1}$.

Hence proved.

Question:62

If $x \sin (a + y) + \sin a \cos (a + y) = 0$, prove that $\frac{d y}{d x}=\frac{\sin ^{2}(a+y)}{\sin a}$.

Answer:

Given that: $x \sin (a+y)+\sin a \cos (a+y)=0$

$\begin{aligned} & \Rightarrow x \sin (a+y)=-\sin a \cos (a+y) \\ & \Rightarrow x=\frac{-\sin a \cdot \cos (a+y)}{\sin (a+y)} \\ & \Rightarrow x=-\sin a \cdot \cot (a+y)\end{aligned}$

Differentiating both sides w.r.t. Y

$\begin{aligned} & \Rightarrow \frac{d x}{d y}=-\sin a \cdot \frac{d}{d y} \cot (a+y) \\ & \Rightarrow \frac{d x}{d y}=-\sin a\left[-\operatorname{cosec}^2(a+y)\right. \\ & \Rightarrow \frac{d x}{d y}=\frac{\sin a}{\sin ^2(a+y)} \\ & \therefore \frac{d y}{d x}=\frac{1}{\frac{d x}{d y}} \\ & =\frac{1}{\frac{\sin a}{\sin ^2(a+y)}}\end{aligned}$

Hence, $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\sin ^2(\mathrm{a}+y)}{\sin \mathrm{a}}$.

Hence proved.

Question:63

If $\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a(x-y)$ prove that $\frac{d y}{d x}=\sqrt{\frac{1-y^{2}}{1-x^{2}}}$

Answer:

Given that: $\sqrt{1-x^2}+\sqrt{1-y^2}=\mathrm{a}(x-y)$

Put $\mathrm{x}=\sin \theta$ and $\mathrm{y}=\sin \Phi$.

$\begin{aligned} & \therefore \theta=\sin ^{-1} \mathrm{x} \text { and } \Phi=\sin ^{-1} \mathrm{y} \\ & \sqrt{1-\sin ^2 \theta}+\sqrt{1-\sin ^2 \phi}=\mathrm{a}(\sin \theta-\sin \Phi) \\ & \Rightarrow \sqrt{\cos ^2 \theta}+\sqrt{\cos ^2 \phi}=\mathrm{a}(\sin \theta-\sin \Phi) \\ & \Rightarrow \cos \theta+\cos \Phi=\mathrm{a}(\sin \theta-\sin \Phi) \\ & \Rightarrow \frac{\cos \theta+\cos \phi}{\sin \theta-\sin \phi}=\mathrm{a}\end{aligned}$

$\begin{aligned} & \Rightarrow \frac{2 \cos \frac{\theta+\phi}{2} \cdot \cos \frac{\theta-\phi}{2}}{2 \cos \frac{\theta+\phi}{2} \cdot \sin \frac{\theta-\phi}{2}}=\mathrm{a} \\ & \Rightarrow \frac{\cos \left(\frac{\theta-\phi}{2}\right)}{\sin \left(\frac{\theta-\phi}{2}\right)}=\mathrm{a} \\ & \Rightarrow \cot \left(\frac{\theta-\phi}{2}\right)=\mathrm{a} \\ & \Rightarrow \frac{\theta-\phi}{2}=\cot ^{-1} \mathrm{a} \\ & \Rightarrow \theta-\Phi=2 \cot ^{-1} \mathrm{a} \\ & \Rightarrow \sin ^{-1} \mathrm{x}-\sin ^{-1} \mathrm{y}=2 \cot ^{-1} \mathrm{a}\end{aligned}$

Differentiating both sides w.r.t. X

$\begin{aligned} & \frac{\mathrm{d}}{\mathrm{dx}}\left(\sin ^{-1} x\right)-\frac{\mathrm{d}}{\mathrm{dx}}\left(\sin ^{-1} x\right)=2 \cdot \frac{\mathrm{~d}}{\mathrm{dx}} \cot ^{-1} \mathrm{a} \\ & \Rightarrow \frac{1}{\sqrt{1-x^2}}-\frac{1}{\sqrt{1-y^2}} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}=0 \\ & \Rightarrow \frac{1}{\sqrt{1-y^2}} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{\sqrt{1-x^2}} \\ & \therefore \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\sqrt{1-y^2}}{\sqrt{1-x^2}} .\end{aligned}$

Question:64 If $y = tan^{-1}x$ , find $\frac{d^2y}{dx^2}$ in terms of y alone.

Answer:

Given that: $\mathrm{y}=\tan ^{-1} \mathrm{x}$

$\Rightarrow x=\tan y$

Differentiating both sides w.r.t. Y

$\frac{d x}{d y}=\sec ^2 y$

$\frac{\mathrm{dy}}{dx}=\frac{1}{\sec ^2 y}=\cos ^2 y$

Again, differentiating both sides w.r.t. x

$\begin{aligned} & \Rightarrow \frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)=\frac{\mathrm{d}}{\mathrm{dx}}\left(\cos ^2 y\right) \\ & \Rightarrow \frac{\mathrm{d}^2 y}{\mathrm{dx}^2}=2 \cos y \cdot \frac{\mathrm{~d}}{\mathrm{dx}}(\cos y) \\ & \Rightarrow \frac{\mathrm{d}^2 y}{\mathrm{dx}^2}=2 \cos y(-\sin y) \cdot \frac{\mathrm{dy}}{\mathrm{dx}} \\ & \Rightarrow \frac{\mathrm{d}^2 y}{\mathrm{dx}^2}=-2 \sin y \cos y \cdot \cos ^2 y \\ & \therefore \frac{\mathrm{~d}^2 y}{\mathrm{dx}^2}=-2 \sin \mathrm{y} \cos ^3 \mathrm{y}\end{aligned}$

Question:65

Verify the Rolle’s theorem for each of the functions
$f(x) = x (x - 1)^2$ in [0, 1].

Answer:

We have, $\mathrm{f}(\mathrm{x})=\mathrm{x}(\mathrm{x}-1)^2$ in $[0,1]$

Since, $f(x)=x(x-1)^2$ is a polynomial function it is continuous in $[0,1]$ and differentiable in $(0,1)$ Now, $f(0)=0$ and $f(1)$

$\Rightarrow \mathrm{f}(0)=\mathrm{f}(1)$

f satisfies the conditions of Rolle's theorem.

Hence, by Rolle's theorem there exists at least one $\mathrm{c} \in(0,1)$ such that $\mathrm{f}^{\prime}(\mathrm{c})=0$

$\begin{aligned} & \Rightarrow 3 \mathrm{c}^2-4 \mathrm{c}+1=0 \\ & \Rightarrow(3 \mathrm{c}-1)(\mathrm{c}-1)=0 \\ & \Rightarrow \mathrm{c}=\frac{1}{3} \in(0,1)\end{aligned}$
Thus, Rolle’s theorem is verified.

Question:66

Verify the Rolle’s theorem for each of the functions
$f(x)=\sin ^{4} x+\cos ^{4} x \text { in }\left[0, \frac{\pi}{2}\right]$

Answer:

We have, $\mathrm{f}(\mathrm{x})=\sin ^4 x+\cos ^4 x$ in $\left[0, \frac{\pi}{2}\right]$

We know that $\sin \mathrm{x}$ and $\cos \mathrm{x}$ are conditions and differentiable

$\therefore \sin 4 x$ and $\cos 4 x$ and hence $\sin 4 x+\cos 4 x$ is continuous and differentiable

Now $\mathrm{f}(0)=0+1=1$ and $\mathrm{f}\left(\frac{\pi}{2}\right)=1+0=1$

$\Rightarrow \mathrm{f}(0)=\mathrm{f}\left(\frac{\pi}{2}\right)$

So, the conditions of Rolle's theorem are satisfied.

Hence, there exists at least one $\mathrm{c} \in\left(0, \frac{\pi}{2}\right)$ such that $\mathrm{f}^{\prime}(\mathrm{c})=0$

$\begin{aligned} & \therefore 4 \sin ^3 \mathrm{c} \cos \mathrm{c}-4 \cos ^3 \mathrm{c} \sin \mathrm{c}=0 \\ & \Rightarrow 4 \sin \mathrm{c} \cos \mathrm{c}\left(\sin ^2 \mathrm{c}-\cos ^2 \mathrm{c}\right)=0 \\ & \Rightarrow 4 \sin \mathrm{c} \cos \mathrm{c}(-\cos 2 \mathrm{c})=0 \\ & \Rightarrow-2 \sin 2 \mathrm{c} \cdot \cos 2 \mathrm{c}=0 \\ & \Rightarrow \sin 4 \mathrm{c}=0 \\ & \Rightarrow 4 \mathrm{c}=\pi \\ & \Rightarrow \mathrm{c}=\frac{\pi}{4} \in\left(0, \frac{\pi}{2}\right)\end{aligned}$

Thus, Rolle’s theorem is verified.

Question:67

Verify Rolle’s theorem for each of the functions
$f(x) = log (x^2 + 2) - log3$ in [- 1, 1].

Answer:

We have, $\mathrm{f}(\mathrm{x})=\log \left(\mathrm{x}^2+2\right)-\log 3$

We know that $\mathrm{x}^2+2$ and the logarithmic function are continuous and differentiable $\therefore \mathrm{f}(\mathrm{x})=\log \left(\mathrm{x}^2+2\right)-\log 3$ is also continuous and differentiable.

Now $f(-1)=f(1)=\log 3-\log 3=0$

So, the conditions of Rolle's theorem are satisfied.

Hence, there exists at least one $\mathrm{c} \in(-1,1)$ such that $\mathrm{f}^{\prime}(\mathrm{c})=0$

$\begin{aligned} & f(x)=\frac{2 c}{c^2+2}-0=0 \\ & \Rightarrow c=0 \in(-1,1)\end{aligned}$

Hence, Rolle's theorem has been verified.

Question:68

Verify Rolle’s theorem for each of the functions
$f(x) = x(x + 3)e^{-x/2}$ in [-3, 0].

Answer:

Given: $f(x) = x(x + 3)e^{-x/2}$
$\Rightarrow f(x)=\left(x^{2}+3 x\right) e^{\frac{-x}{2}}$
Now, we have to show that f(x) verifies Rolle’s Theorem
First of all, the Conditions of Rolle’s theorem are:
a) f(x) is continuous at (a,b)
b) f(x) is derivable at (a,b)
c) f(a) = f(b)
If all three conditions are satisfied, then there exists some ‘c’ in (a,b) such that f’(c) = 0
Condition 1:
$\Rightarrow f(x)=\left(x^{2}+3 x\right) e^{\frac{-x}{2}}$
Since f(x) is the multiplication of algebra and exponential function,n and is defined everywhere in its domain.
$\Rightarrow f(x)=\left(x^{2}+3 x\right) e^{\frac{-x}{2}}$ is continuous at x ∈ [-3,0]
Hence, condition 1 is satisfied.
Condition 2: $\Rightarrow f(x)=\left(x^{2}+3 x\right) e^{\frac{-x}{2}}$
On differentiating f(x) concerning x, we get,

$\\ f^{\prime}(x)=e^{\frac{-x}{2}} \frac{d}{d x}\left(x^{2}+3 x\right)+\left(x^{2}+3 x\right) \frac{d}{d x} e^{-\frac{x}{2}} \text { [by product rule }] \\ \Rightarrow f^{\prime}(x)=e^{-\frac{x}{2}}[2 x+3]+\left(x^{2}+3 x\right) \times\left(-\frac{1}{2} e^{-\frac{x}{2}}\right) \\ \Rightarrow f^{\prime}(x)=2 x e^{-\frac{x}{2}}+3 e^{-\frac{x}{2}}-\frac{x^{2}}{2} e^{-\frac{x}{2}}-\frac{3 x}{2} \mathrm{e}^{-\frac{x}{2}} \\ \Rightarrow f^{\prime}(x)=e^{-\frac{x}{2}}\left[2 x+3-\frac{x^{2}}{2}-\frac{3 x}{2}\right] \\ \Rightarrow f^{\prime}(x)=\frac{e^{-\frac{x}{2}}}{2}\left[x+6-x^{2}\right] \\ \Rightarrow f^{\prime}(x)=-\frac{e^{-\frac{x}{2}}}{2}\left[x^{2}-x-6\right] \\ \Rightarrow \quad(x)=-\frac{e^{-\frac{x}{2}}}{2}\left[x^{2}-3 x+2 x-6\right]$
$\\ \Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=-\frac{\mathrm{e}^{-\frac{\mathrm{x}}{2}}}{2}[\mathrm{x}(\mathrm{x}-3)+2(\mathrm{x}-3)] \\ \Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=-\frac{\mathrm{e}^{-\frac{\mathrm{x}}{2}}}{2}[(\mathrm{x}-3)(\mathrm{x}+2)]$
⇒ f(x) is differentiable at [-3,0]
Hence, condition 2 is satisfied.
Condition 3:
$\\f(x)=\left(x^{2}+3 x\right) e^{\frac{-x}{2}}$ \\$f(-3)=\left[(-3)^{2}+3(-3)\right] e^{\frac{-(-3)}{2}}$ \\$=[9-9] \mathrm{e}^{2 / 2}$ \\$=0$ \\$f(0)=\left[(0)^{2}+3(0)\right] e^{\frac{-0}{2}}$ \\$=0$ \\Hence, $f(-3)=f(0)$ \\Hence, condition 3 is also satisfied. \\Now, let us show that $c \in(0,1)$ such that $f^{\prime}(c)=0$ \\$f(x)=\left(x^{2}+3 x\right) e^{\frac{-x}{2}}$
$$On differentiating above with respect to x , we get $\\ f^{\prime}(x)=-\frac{e^{-\frac{x}{2}}}{2}[(x-3)(x+2)]$
$\\ Put$ x=c in above equation, we get$\\ \mathrm{f}^{\prime}(\mathrm{c})=-\frac{\mathrm{e}^{-\frac{\mathrm{c}}{2}}}{2}[(\mathrm{c}-3)(\mathrm{c}+2)]$
All three conditions of Rolle’s theorem are satisfied
f’(c) = 0
$\begin{aligned} &-\frac{e^{-\frac{c}{2}}}{2}[(c-3)(c+2)]=0\\ &\because-\frac{e^{-\frac{c}{2}}}{2} c a n^{\prime} t \text { be zero }\\ &\Rightarrow(c-3)(c+2)=0\\ &\Rightarrow c-3=0 \text { or } c+2=0\\ &\Rightarrow c=3 \text { or } c=-2\\ &\text { So, value of } c=-2,3\\ &c=-2 \in(-3,0) \text { but } c=3 \in(-3,0)\\ &\therefore c=-2 \end{aligned}$
Thus, Rolle’s theorem is verified.

Question:69 Verify Rolle’s theorem for each of the functions
$f(x)=\sqrt{4-x^{2}} \text { in }[-2,2]$

Answer:

We have, $\sqrt{4-x^2}=\left(4-x^2\right)^{\frac{1}{2}}$

Since $\left(4-x^2\right)$ and the square root function are continuous and differentiable in their domain, given function $f(x)$ is also continuous and differentiable in $[-2,2]$

Also $f(-2)=f(2)=0$

So, the conditions of Rolle's theorem are satisfied.

Hence, there exists a real number $\mathrm{c} \in(-2,2)$ such that $\mathrm{f}^{\prime}(\mathrm{c})=0$.

Now $f^{\prime}(\mathrm{x})=\frac{1}{2}\left(4-x^2\right)^{\frac{-1}{2}}(-2 x)$

$=-\frac{x}{\sqrt{4-x^2}}$

So, $f^{\prime}(c)=0$

$\begin{aligned} & \Rightarrow \frac{c}{\sqrt{4-c^2}}=0 \\ & \Rightarrow c=0 \in(-2,2)\end{aligned}$

Hence, Rolle's theorem has been verified.

Question:70

Discuss the applicability of Rolle’s theorem on the function given by
$f(x)=\begin{array}{ll} x^{2}+1, & \text { if } 0 \leq x \leq 1 \\ 3-x, & \text { if } 1 \leq x \leq 2 \end{array}$

Answer:

Given: $f(x)=\begin{array}{ll} x^{2}+1, & \text { if } 0 \leq x \leq 1 \\ 3-x, & \text { if } 1 \leq x \leq 2 \end{array}$
First of all, the Conditions of Rolle’s theorem are:
a) f(x) is continuous at (a,b)
b) f(x) is derivable at (a,b)
c) f(a) = f(b)
If all three conditions are satisfied, then there exists some ‘c’ in (a,b) such that f’(c) = 0
Condition 1:
At x = 1
$\\\lim _{\mathrm{LHL}}\left(\mathrm{x}^{2}+1\right)=1+1=2$ \\$\lim _{\mathrm{RHL}}=\lim _{x \rightarrow 1^{+}}(3-\mathrm{x})=3-1=2$ \\$\because \mathrm{LHL}=\mathrm{RHL}=2$
and $f(1)=3-x=3-1=2$
$\therefore f(x)$ is continuous at $x=1$
Hence, condition 1 is satisfied.
Condition 2:
Now, we have to check f(x) is differentiable
$f(x)=\left\{\begin{array}{l}x^{2}+1, \text { if } 0 \leq x \leq 1 \\ 3-x, \text { if } 1 \leq x \leq 2\end{array}\right.$
On differentiating concerning x, we get
$\Rightarrow f^{\prime}(x)=\left\{\begin{array}{c}2 x+0, \text { if } 0<x<1 \\ 0-1, \text { if } 1<x<2\end{array}\right.$ or
$\mathrm{f}^{\prime}(\mathrm{x})=\left\{\begin{array}{l} 2 \mathrm{x}, \text { if } 0<\mathrm{x}<1 \\ -1, \text { if } 1<\mathrm{x}<2 \end{array}\right.$
Now, let us consider the differentiability of f(x) at x = 1
LHD ⇒ f(x) = 2x = 2(1) = 2
RHD ⇒ f(x) = -1 = -1
LHD ≠ RHD
∴ f(x) is not differentiable at x = 1
Thus, Rolle’s theorem does not apply to the given function.

Question:71

Find the points on the curve y = (cosx - 1) in [0, $2\pi$], where the tangent is parallel to the x-axis.

Answer:

We have $y=\cos x-1$

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}}=-\sin \mathrm{x}$

For the tangent to be parallel to the x-axis

We must have $\frac{d y}{d x}=0$

$\begin{aligned} & \therefore-\sin \mathrm{x}=0 \\ & \therefore \mathrm{x}=\pi \in[0,2 \pi]\end{aligned}$

$y(\pi)=\cos \pi-1=-2$

Hence, the required point on the curve, where the tangent drawn is parallel to the x-axis $(\pi,-2)$

Question:72

Using Rolle’s theorem, find the point on the curve y = x(x - 4), $x\in [0,4]$ where the tangent is parallel to x-the axis.

Answer:

We have, $y=x(x-4), x \in[0,4]$

Since the given function is polynomial, it is continuous and differentiable.

Also $y(0)=y(4)=0$

So, the conditions of Role's theorem are satisfied.

Hence there exists a point $\mathrm{c} \in(0,4)$ such that $\mathrm{f}^{\prime}(\mathrm{c})=0$

$\Rightarrow 2 c-4=0$

$\Rightarrow c=2$

$\Rightarrow x=2$ and $y(2)$

$\begin{aligned} & =2(2-4) \\ & =-4\end{aligned}$

Therefore, the required point on the curve, where the tangent drawn is parallel to the x-axis is $(2,-4)$.

Question:73

Verify the mean value theorem for each of the functions given
$f(x)=\frac{1}{4 x-1} \text { in }[1,4]$

Answer:

We have, $\mathrm{f}(\mathrm{x})=\frac{1}{4 x-1}$ in $[1,4]$

Clearly $f(x)$ is continuous in $[1,4]$

Also, $\mathrm{f}^{\prime}(\mathrm{x})=-\frac{4}{(4 x-1)^2}$, which exists in $(1,4)$

So, it is differentiable in $(1,4)$

Thus conditions of the mean value theorem are satisfied.

Hence, there exists a real number $\mathrm{c} \in(1,4)$ such that

$\begin{aligned} & f(\mathrm{c})=\frac{\mathrm{f}(4)-f(1)}{4-1} \\ & \Rightarrow \frac{-4}{(4 \mathrm{c}-1)^2}=\frac{\frac{1}{16-1}-\frac{1}{4-1}}{4-1} \\ & =\frac{\frac{1}{15}-\frac{1}{3}}{3} \\ & \Rightarrow \frac{-4}{(4-1)^2}=\frac{-4}{45} \\ & \Rightarrow(4 \mathrm{c}-1)^2=45 \\ & \Rightarrow 4 \mathrm{c}-1= \pm 3 \sqrt{5} \\ & \Rightarrow \mathrm{c}=\frac{3 \sqrt{5}+1}{4} \in(1,4)\end{aligned}$

Hence mean value theorem has been verified.

Question:74

Verify the mean value theorem for each of the functions given
$f(x) = x^3 - 2x^2 - x + 3 $ in [0, 1]$

Answer:

We have, $f(x)=x^3-2 x^2-x+3$ in $[0,1]$

Since, $f(x)$ is a polynomial function it is continuous in $[0,1]$ and differentiable in $(0,1)$

Thus, the conditions of the mean value theorem are satisfied.

Hence, there exists a real number $\mathrm{c} \in(0,1)$ such that

$\begin{aligned} & f(\mathrm{c})=\frac{\mathrm{f}(1)-\mathrm{f}(0)}{1-0} \\ & \Rightarrow 3 \mathrm{c}^2-4 \mathrm{c}-1=\frac{[1-2-1+3]-[0+3]}{1-0} \\ & \Rightarrow 3 \mathrm{c}^2-4 \mathrm{c}-1=-2 \\ & \Rightarrow 3 \mathrm{c}^2-4 \mathrm{c}+1=0 \\ & \Rightarrow(3 \mathrm{c}-1)(\mathrm{c}-1)=0 \\ & \Rightarrow \mathrm{c}=\frac{1}{3} \in(0,1)\end{aligned}$

Hence, the mean value theorem has been verified.

Question:75

Verify the mean value theorem for each of the functions given
$f(x) = sinx - sin2x$ in $[0,\pi]$

Answer:

We have, $\mathrm{f}(\mathrm{x})=\sin \mathrm{x}-\sin 2 \mathrm{x}$ in $[0, \pi]$

We know that all trigonometric functions are continuous and differentiable in their domain, a given function is also continuous and differentiable

So, the condition of the mean value theorem is satisfied.

Hence, there exists at least one $\mathrm{c} \in(0, \pi)$ such that,

$\begin{aligned} & \mathrm{f}^{\prime}(\mathrm{c})=\frac{\mathrm{f}(\pi)-\mathrm{f}(0)}{\pi-0} \\ & \Rightarrow \cos \mathrm{c}-2 \cos 2 \mathrm{c}=\frac{\sin \pi-\sin 2 \pi-\sin 0+\sin 0}{\pi-0} \\ & \Rightarrow 2 \cos 2 \mathrm{c}-\cos \mathrm{c}=0 \\ & \Rightarrow 2\left(2 \cos ^2 \mathrm{c}-1\right)-\cos \mathrm{c}=0 \\ & \Rightarrow 4 \cos ^2 \mathrm{c}-\cos \mathrm{c}-2=0 \\ & \Rightarrow \cos \mathrm{c}=\frac{1 \pm \sqrt{1+32}}{8} \\ & =\frac{1 \pm \sqrt{33}}{8} \\ & \Rightarrow \mathrm{c}=\cos ^{-1}\left(\frac{1 \pm \sqrt{33}}{8}\right) \in(0, \pi)\end{aligned}$

Hence, the mean value theorem has been verified.

Question:76

Verify the mean value theorem for each of the functions given
$f(x)=\sqrt{25-x^2}$ in [1,5]

Answer:

We have, $\mathrm{f}(\mathrm{x})=\sqrt{25-x^2}$ in $[1,5]$

Since $25-x^2$ and the square root function are continuous and differentiable in their domain, given function $f(x)$ is also continuous and differentiable.

So, the condition of the mean value theorem is satisfied.

Hence, there exists at least one $\mathrm{c} \in(1,5)$ such that,

$\begin{aligned} & \mathrm{f}^{\prime}(\mathrm{c})=\frac{\mathrm{f}(5)-\mathrm{f}(1)}{5-1} \\ & \Rightarrow \frac{-\mathrm{c}}{\sqrt{25-\mathrm{c}^2}}=\frac{0-\sqrt{24}}{4} \\ & \Rightarrow 16 \mathrm{c}^2=24\left(25-\mathrm{c}^2\right) \\ & \Rightarrow 40 \mathrm{c}^2=600 \\ & \Rightarrow \mathrm{c}^2=15 \\ & \Rightarrow \mathrm{c}=\sqrt{15} \in(1,5)\end{aligned}$

Hence, the mean value theorem has been verified.

Question:77

Find a point on the curve $y = (x - 3)^2$, where the tangent is parallel to the chord joining the points (3, 0) and (4, 1).

Answer:

We have, $\mathrm{y}=(\mathrm{x}-3)^2$, which is polynomial function.

So it is continuous and differentiable.

Thus conditions of the mean value theorem are satisfied.

Hence, there exists at least one $\mathrm{c} \in(3,4)$ such that,

$\begin{aligned} & \mathrm{f}^{\prime}(\mathrm{c})=\frac{\mathrm{f}(4)-\mathrm{f}(3)}{4-3} \\ & \Rightarrow 2(\mathrm{c}-3)=\frac{1-0}{1} \\ & \Rightarrow \mathrm{c}-3=\frac{1}{2} \\ & \Rightarrow \mathrm{c}=\frac{7}{2} \in(3,4)\end{aligned}$

$\Rightarrow \mathrm{x}=\frac{7}{2}$, where tangent is parallel to the chord joining points $(3,0)$ and $(4,1)$.

For $\mathrm{x}=\frac{7}{2}, \mathrm{y}=\left(\frac{7}{2}-3\right)^2$

$\begin{aligned} & =\left(\frac{1}{2}\right)^2 \\ & =\frac{1}{4}\end{aligned}$

So, $\left(\frac{7}{2}, \frac{1}{4}\right)$ is the point on the curve, where the tangent drawn is parallel to the chord joining the points $(3,0)$ and $(4,1)$.

Question:78

Using the mean value theorem, prove that there is a point on the curve $y = 2x^2 - 5x + 3$ between the points A(1, 0) and B(2, 1), where the tangent is parallel to the chord AB. Also, find that point.

Answer:

We have, $\mathrm{y}=2 \mathrm{x}^2-5 \mathrm{x}+3$, which is polynomial function.

So it is continuous and differentiable.

Thus conditions of the mean value theorem are satisfied.

Hence, there exists at least one $\mathrm{c} \in(1,2)$ such that,

$\begin{aligned} & \mathrm{f}(\mathrm{c})=\frac{\mathrm{f}(2)-\mathrm{f}(1)}{2-1} \\ & \Rightarrow 4 \mathrm{c}-5=\frac{1-0}{1} \\ & \Rightarrow 4 \mathrm{c}-5=1 \\ & \therefore \mathrm{c}=\frac{3}{2} \in(1,2)\end{aligned}$

For $\mathrm{x}=\frac{3}{2}, \mathrm{y}=2\left(\frac{3}{2}\right)^2-5\left(\frac{3}{2}\right)+3=0$

Hence, $\left(\frac{3}{2}, 0\right)$ is the points on the curve $\mathrm{y}=2 \mathrm{x}^2-5 \mathrm{x}+3$ between the points $\mathrm{A}(1,0)$ and $\mathrm{B}(2,1)$, where tangent is parallel to the chord AB .

Question:79

Find the values of p and q so that
$f(x)=\left\{\begin{array}{cl} x^{2}+3 x+p, & \text { if } x \leq 1 \\ q x+2, & \text { if } x>1 \end{array}\right.$
It is differentiable at x = 1.

Answer:

Given that,
$f(x)=\left\{\begin{array}{cl} x^{2}+3 x+p, & \text { if } x \leq 1 \\ q x+2, & \text { if } x>1 \end{array}\right.$
It is differentiable at x = 1.
We know that, f(x) is differentiable at x =1 ⇔ Lf’(1) = Rf’(1).

$\\ {L f^{\prime}(1)}=\lim _{x \rightarrow 1^{-}} \frac{f(x)-f(1)}{x-1} \\ =\lim _{h \rightarrow 0} \frac{f(1-h)-f(1)}{(1-h)-1} \\ =\lim _{h \rightarrow 0} \frac{\left[\left\{(1-h)^{2}+3(1-h)+p\right]-(1+3+p)\right]}{(1-h)-1} \quad\left(\because f(x)=x^{2}+3 x+p, \text { if } x \leq 1\right) \\ =\lim _{h \rightarrow 0} \frac{\left[\left(1+h^{2}-2 h+3-3 h+p\right)-(4+p)\right]}{-h} \\ =h m \frac{\left[h^{2}-5 h+p+4-4-p\right]}{-h} \\ = \quad \lim _{h \rightarrow 0} \frac{\left[h^{2}-5 h\right]}{-h}=\lim _{h \rightarrow 0} \frac{h(h-5)}{-h} \\ =\lim _{h \rightarrow 0} h(5-h)$
$\\ =5 \\ \lim _{\operatorname{Rf}^{\prime}(1)}=\lim _{x \rightarrow 1} \frac{f(x)-f(1)}{x-1} \\ =\lim _{h \rightarrow 0} \frac{f(1+h)-f(1)}{(1+h)-1} \\ =\lim _{h \rightarrow 0} \frac{[(q(1+h)+2)-(q+2)]}{(1+h)-1} \quad(v f(x)=q x+2, \text { if } x>1) \\ =\lim _{h \rightarrow 0} \frac{[(q+q h+2)-(q+2)]}{h} \\ =\lim _{h \rightarrow 0} \frac{[q+q h+2-q-2]}{h} \\ =q$
Since, Lf’(1) = Rf’(1)
∴ 5 = q (i)
Now, we know that if a function is differentiable at a point, it is necessarily continuous at that point.
⇒ f(x) is continuous at x = 1.
⇒ f(1-) = f(1+) = f(1)
⇒ 1+3+p = q+2 = 1+3+p
⇒ p-q = 2-4 = -2
⇒ q-p = 2
Now substituting the value of ‘q’ from (i), we get
⇒ 5-p = 2
⇒ p = 3
∴ p = 3 and q = 5

Question:80

A.If $x^m.y^n = (x+y)^{m+n}$ prove that $\frac{dy}{dx}= \frac{y}{x}$
B. If $x^m.y^n = (x+y)^{m+n}$ prove that $\frac{d^2y}{dx^2}=0$

Answer:

A. We have,
$x^m.y^n = (x+y)^{m+n}$
Taking logs on both sides, we get

$\log \left(x^m y^n\right)=\log (x+y)^{m+n} \Rightarrow m \log x+n \log y=(m+n) \log (x+y)$

Differentiating both sides w.r.t x, we get

$\begin{aligned} & \mathrm{m} \cdot \frac{1}{\mathrm{x}}+\mathrm{n} \cdot \frac{1}{\mathrm{y}} \frac{\mathrm{dy}}{\mathrm{dx}}=(\mathrm{m}+\mathrm{n}) \frac{1}{\mathrm{x}+\mathrm{y}} \cdot \frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{x}+\mathrm{y})=\frac{m}{x}+\frac{n}{y} \frac{d y}{d x}=\frac{m+n}{x+y} \cdot\left(1+\frac{d y}{d x}\right) \\ & \Rightarrow\left(\frac{n}{y}-\frac{m+n}{x+y}\right) \frac{d y}{d x}=\frac{m+n}{x+y}-\frac{m}{x} \\ & \Rightarrow\left(\frac{n x+n y-m y-n y}{y(x+y)}\right) \frac{d y}{d x}=\left(\frac{m x+n x-m x-m y}{x(x+y)}\right) \\ & \Rightarrow\left(\frac{n x-m y}{y(x+y)}\right) \frac{d y}{d x}=\left(\frac{n x-m x}{x(x+y)}\right) \\ & \Rightarrow \frac{d y}{d x}=\left(\frac{n x-m x}{x(x+y)}\right)\left(\frac{y(x+y)}{n x-m y}\right) \\ & \Rightarrow \frac{d y}{d x}=\frac{y}{x}\end{aligned}$

Hence proved.

B. We have,
$\begin{aligned} &\frac{d y}{d x}=\frac{y}{x}\\ &\text { Differentiating both sides w.r.t } x, \text { we get }\\ &\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\frac{\mathrm{x} \frac{\mathrm{dy}}{\mathrm{dx}}-\mathrm{y} \cdot 1}{\mathrm{x}^{2}}\\ &\frac{\mathrm{d}^{2} y}{\mathrm{dx}^{2}}=\frac{\mathrm{x} \frac{\mathrm{y}}{\mathrm{x}}-\mathrm{y} \cdot \mathrm{.}}{\mathrm{x}^{2}}\left(\because \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{y}}{\mathrm{x}}\right)\\ &\frac{d^{2} y}{d x^{2}}=\frac{y-y}{x^{2}}=0\\ &\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=0 \end{aligned}$
Hence Proved.

Question:81

If x = sin t and y = sin pt, prove that

$(1-x^2)$$\frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}+p^{2} y=0$

Answer:

We have,
$\begin{aligned} &x=\sin t \text { and } y=\sin p t\\ &\Rightarrow \frac{\mathrm{dx}}{\mathrm{dt}}=\cos \mathrm{t} \text { and } \frac{\mathrm{dy}}{\mathrm{dt}}=\mathrm{p} \cos \mathrm{pt}\\ &\therefore \frac{\mathrm{d} y}{\mathrm{dx}}=\frac{\mathrm{dy} / \mathrm{dt}}{\mathrm{dx} / \mathrm{dt}}=\frac{\mathrm{p} \cdot \cos \mathrm{pt}}{\cos \mathrm{t}}\\ &\Rightarrow \frac{d y}{d x}=\frac{p \cdot \cos p t}{\cos t}\\ &\text { Differentiating both sides w.r.t } x, \text { we get }\\ &\frac{\mathrm{d}^{2 y}}{\mathrm{dx}^{2}}=\frac{\operatorname{cost} \frac{\mathrm{d}}{\mathrm{dt}}(\mathrm{p} \cdot \cos \mathrm{pt}) \frac{\mathrm{dt}}{\mathrm{dx}}-\mathrm{p} \cdot \cos \mathrm{pt} \frac{\mathrm{d}}{\mathrm{dt}} \operatorname{cost} \frac{\mathrm{dt}}{\mathrm{dx}}}{\cos ^{2} \mathrm{t}}\\ \end{aligned}$
$\begin{aligned} &\Rightarrow \frac{d^{2 y}}{d x^{2}}=\frac{\left(-p^{2} \sin p t \cos t+p \cdot s i n t \cos p t\right) \frac{1}{\operatorname{cost}}}{\cos ^{2} t}\left(\because \frac{d x}{d t}=\cos t \Rightarrow \frac{d t}{d x}=\frac{1}{\operatorname{cost}}\right)\\ &\Rightarrow \frac{\mathrm{d}^{2 y}}{\mathrm{dx}^{2}}=\frac{-\mathrm{p}^{2} \sin \mathrm{pt} \cdot \cos t+\mathrm{p} \cdot \operatorname{sint} \cos \mathrm{pt}}{\left(1-\sin ^{2} \mathrm{t}\right) \operatorname{cost}}\\ &\Rightarrow\left(1-\sin ^{2} t\right) \frac{d^{2 y}}{d x^{2}}=\frac{-p^{2} \sin p t \cdot \cos t+p \sin t \cos p t}{\operatorname{cost}}\\ &\Rightarrow\left(1-\sin ^{2} t\right) \frac{d^{2 y}}{d x^{2}}=\frac{-p^{2} \sin p t \cdot \cos t}{\cos t}+\frac{p \cdot s i n t \cos p t}{\operatorname{cost}}\\ \end{aligned}$
$\\ \Rightarrow\left(1-x^{2}\right) \frac{d^{2 y}}{d x^{2}}=-p^{2} y+x \frac{d y}{d x}\left(\because x=\operatorname{sint}, y=\sin p t \text { and } \frac{d y}{d x}=\frac{p \cdot \cos p t}{\cos t}\right)\\ \\\Rightarrow\left(1-x^{2}\right) \frac{d^{2 y}}{d x^{2}}-x \frac{d y}{d x}+p^{2} y=0$
Hence proved.

Question:82

Find $\frac{d y}{d x}, \text { if } y=x^{\tan x}+\sqrt{\frac{x^{2}+1}{2}}$

Answer:

We have,
$y=x^{\tan x}+\sqrt{\frac{x^{2}+1}{2}}$
Putting $x^{\tan x}=u$ and $\sqrt{\frac{x^2+1}{2}}=v$

$u=x^{\tan x}$

Taking logs on both sides, we get

$\log u=\tan x \log x$

Differentiating w.r.t x , we get

$\begin{aligned} & \Rightarrow \frac{1}{u} \frac{d u}{d x}=\tan x \cdot \frac{1}{x}+\log x \cdot \sec ^2 x \\ & \Rightarrow \frac{d u}{d x}=u\left(\tan x \cdot \frac{1}{x}+\log x \cdot \sec ^2 x\right) \\ & \Rightarrow \frac{d u}{d x}=x^{\tan x}\left(\tan x \cdot \frac{1}{x}+\log x \cdot \sec ^2 x\right)\end{aligned}$

Now, $\mathrm{v}=\sqrt{\frac{\mathrm{x}^2+1}{2}}$

$\Rightarrow v=\left(\frac{x^2+1}{2}\right)^{1 / 2}$

Differentiating w.r.t x, we get

$\Rightarrow \frac{d v}{d x}=\frac{1}{2}\left(\frac{x^2+1}{2}\right)^{-1 / 2} \cdot \frac{2 x}{2} \Rightarrow \frac{d v}{d x}=\frac{x}{2}\left(\frac{2}{x^2+1}\right)^{1 / 2}=\frac{x}{2} \sqrt{\frac{2}{x^2+1}}$

Now, $y=u+v$

$\begin{aligned} & \Rightarrow \frac{d y}{d x}=\frac{d u}{d x}+\frac{d v}{d x} \\ & \Rightarrow \frac{d y}{d x}=x^{\tan x}\left(\tan x \cdot \frac{1}{x}+\log x \cdot \sec ^2 x\right)+\frac{x}{2} \sqrt{\frac{2}{x^2+1}} \\ & \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{x}^{\tan \mathrm{x}}\left(\frac{\tan \mathrm{x}}{\mathrm{x}}+\log \mathrm{x} \cdot \sec ^2 \mathrm{x}\right)+\frac{\mathrm{x}}{\sqrt{2\left(\mathrm{x}^2+1\right)}}\end{aligned}$

Question:83

If f(x) = 2x and g(x) = $\frac{x^2}{2}+1$ then which of the following can be a discontinuous function?
A. f(x) + g(x)
B. f(x) - g(x)
C. f(x). g(x)
D. $\frac{g(x)}{f(x)}$

Answer:

We know that if two functions f(x) and g(x) are continuous then {f(x) +g(x)},{f(x)-g(x)}, {f(x).g(x)} and $\left\{\left(\frac{\mathrm{f}(\mathrm{x})}{\mathrm{g}(\mathrm{x})}\right), \text { when } \mathrm{g}(\mathrm{x}) \neq 0\right\}$ are continuous.
Since, f(x) = 2x and $g(x)=$$\frac{x^2}{2}+1$ are polynomial functions, they are continuous everywhere.
⇒ {f(x) +g(x)},{f(x)-g(x)}, {f(x).g(x)} are continuous functions.
for, $\frac{g(x)}{f(x)}=\frac{\frac{x^{2}}{2}+1}{2 x}=\frac{x^{2}+2}{4 x}$
now, f(x) = 0
⇒ 4x = 0
⇒ x = 0
∴ $\frac{g(x)}{f(x)}$ is discontinuous at x=0.

Hence, the correct answer is option (D).

Question:90

Let $f(x) = |sinx|$. Then
A. f is everywhere differentiable
B. f is everywhere continuous but not differentiable $x = n\pi,n\in Z$
C. f is everywhere continuous but not differentiable at $x = (2n+1)\frac{\pi}{2},n\in Z$
D. None of these

Answer:
Given that, $f(x) = |sinx|$
Let g(x) = sinx and h(x) = |x|
Then, f(x) = hog(x)
We know that the modulus function and sine function are continuous everywhere. Since
The imposition of two continuous functions is a continuous function.
Hence, f(x) = hog(x) is continuous everywhere.
Now, v(x)=|x| is not differentiable at x = 0.
$\begin{aligned} &LHD=\lim _{0^{-}} \frac{\mathrm{v}(\mathrm{x})-\mathrm{v}(0)}{\mathrm{x}-0}\\ &\lim _{h \rightarrow 0} \frac{v(0-h)-v(0)}{(0-h)-0}\\ &\lim _{=h \rightarrow 0} \frac{|0-h|-|0|}{-h} \quad(\because v(x)=|x|)\\ &\lim _{h \rightarrow 0} \frac{|-h|}{-h}\\ &=\lim _{h \rightarrow 0} \frac{h}{-h}\\ &=\lim _{h \rightarrow 0}-1=-1\\ \end{aligned}$
$R H D=\lim _{h \rightarrow 0} \frac{v(0+h)-v(0)}{(0+h)-0}$

$=\lim _{h \rightarrow 0} \frac{|0+h|-|0|}{h} \quad(\because v(x)=|x|) \lim _{h \rightarrow 0} \frac{|h|}{h}=\lim _{h \rightarrow 0} \frac{h}{h}=\lim _{h \rightarrow 0} 1=1 \Rightarrow \mathrm{LY}^{\prime}(0) \neq \mathrm{RC}^{\prime}(0) \Rightarrow|\mathrm{x}|$ is not differentiable at $\mathrm{x}=0 . \Rightarrow \mathrm{h}(\mathrm{x})$ is not differentiable at $\mathrm{x}=0$.

So, $\mathrm{f}(\mathrm{x})$ is not differentiable where $\sin x=0$

We know that $\sin x=0$ at $x=n \pi, n \in Z$

Hence, $\mathrm{f}(\mathrm{x})$ is everywhere continuous but not differentiable $x=\mathrm{n} \pi, \mathrm{n} \in \mathrm{Z}$.

Hence, the correct answer is option (B).

Question:91

If $y=\log \left(\frac{1-x^{2}}{1+x^{2}}\right)$ then $\frac{dy}{dx}$ is equal to

$\\A.\frac{4 x^{3}}{1-x^{2}}$ \B.$\frac{-4 x}{1-x^{4}}$ \C. $\frac{1}{4-\mathrm{x}^{4}}$ \D. $\frac{-4 x^{3}}{1-x^{4}}$$

Answer:

$\begin{aligned} &\text { We have, }\\ &y=\log \frac{1-x^{2}}{1+x^{2}}\\ &\Rightarrow \mathrm{y}=\log \left(1-\mathrm{x}^{2}\right)-\log \left(1+\mathrm{x}^{2}\right)\\ &\Rightarrow \frac{d y}{d x}=\frac{1}{1-x^{2}} \frac{d}{d x}\left(1-x^{2}\right)-\frac{1}{1+x^{2}} \frac{d}{d x}\left(1+x^{2}\right)\\ &\Rightarrow \frac{d y}{d x}=\frac{1}{1-x^{2}} \cdot(-2 x)-\frac{1}{1+x^{2}} \cdot(2 x)\\ &\Rightarrow \frac{d y}{d x}=\frac{-2 x}{1-x^{2}}-\frac{2 x}{1+x^{2}}\\ &\frac{d y}{d x}=\frac{-2 x\left(1+x^{2}\right)-2 x\left(1-x^{2}\right)}{\left(1-x^{2}\right)\left(1+x^{2}\right)}\\ &\frac{d y}{d x}=\frac{-2 x-2 x^{3}-2 x+2 x^{3}}{\left(1-x^{4}\right)}\\ &\frac{d y}{d x}=\frac{-4 x}{\left(1-x^{4}\right)} \end{aligned}$

Hence, the correct answer is option (B).

Question:92

If $y=\sqrt{\sin x+y}$ then $\frac{d y}{d x}$ is equal to

$\\A. \ \frac{\cos \mathrm{x}}{2 \mathrm{y}-1}$ \\\\B. $\frac{\cos x}{1-2 y}$ \\\\C. $\frac{\sin \mathrm{x}}{1-2 \mathrm{y}}$\\\\ D. $\frac{\sin \mathrm{x}}{2 \mathrm{y}-1}$$

Answer:
$\mathrm{y}=\sqrt{\sin \mathrm{x}+y}$

Squaring both sides, we get

$y^2=\sin x+y$

Differentiating w.r.t y, we get

$\begin{aligned} & 2 y=\cos x \frac{d x}{d y}+1 \\ & \frac{d x}{d y}=\frac{2 y-1}{\cos x} \\ & \therefore \frac{d y}{d x}=\frac{\cos x}{2 y-1}\end{aligned}$

Hence, the correct answer is option (A).

Question:93

The derivative of $\cos ^{-1}\left(2 x^{2}-1\right)$ w.r.t $\cos ^{-1} x$ is

$\begin{aligned} & A 2 \\ & B \frac{-1}{2 \sqrt{1-\mathrm{x}^2}} \\ & C \frac{2}{x} \\ & D .1-x^2\end{aligned}$

Answer:

Let $u=\cos ^{-1}\left(2 x^2-1\right)$ and $v=\cos ^{-1} x$

Now, $u=\cos ^{-1}\left(2 x^2-1\right)$

$\begin{aligned} & u=\cos ^{-1}\left(2 \cos ^2 v-1\right)\left[\cdot v=\cos ^{-1} x \rightarrow \cos v=x\right] \\ & u=\cos ^{-1}(\cos 2 v)\left[\because 2 \cos ^2 x-1=\cos 2 x\right] \\ & \Rightarrow u=2 v \\ & \frac{d u}{d v}=2\end{aligned}$

Hence, the correct answer is option (A).

Question:94

If $x = t^2, y = t^3$, then $\frac{d^2y}{dx^2}$ is
A. $\frac{3}{2}$
B. $\frac{3}{4t}$
C. $\frac{3}{2t}$
D. $\frac{3}{4}$

Answer:
$\begin{aligned} &\text { Given that, } x=t ^2, y=t^{3}\\ &\Rightarrow \frac{d x}{d t}=2 t \text { and } \frac{d y}{d t}=3 t^{2}\\ & \frac{d y}{d x}=\frac{d y / d t}{d x / d t}\\ &\frac{d y}{d x}=\frac{3 t^{2}}{2 t}=\frac{3 t}{2}\\ &\frac{d^{2} y}{d x^{2}}=\frac{3}{2} \frac{d t}{d x}\\ &\frac{d^{2} y}{d x^{2}}=\frac{3}{2} \cdot \frac{1}{2 t}\left(\because \frac{d x}{d t}=2 t \Rightarrow \frac{d t}{d x}=\frac{1}{2 t}\right)\\ &\frac{d^{2} y}{d x^{2}}=\frac{3}{4 t} \end{aligned}$

Hence, the correct answer is option (B).

Question:95

The value of c in Rolle’s theorem for the function $f(x) = x^3 - 3x$ in the interval $[0,\sqrt 3]$ is
A. 1
B.-1
C. $\frac{3}{2}$
D.$\frac{1}{3}$

Answer:

The given function is $f(x)=x^3-3 x$.

This is a polynomial function, which is continuous and derivable in $R$.

Therefore, the function is continuous on $[0, \sqrt{3}]$ and derivable on $(0, \sqrt{3})$

Differentiating the given function with respect to $x$, we get

$\begin{aligned} & f^{\prime}(x)=3 x^2-3 \\ & \Rightarrow f^{\prime}(c)=3 c^2-3 \\ & \therefore f^{\prime}(c)=0 \\ & \Rightarrow 3 c^2-3=0 \\ & \Rightarrow c^2=1 \\ & \Rightarrow c= \pm 1\end{aligned}$

Thus, $c=1 \in[0, \sqrt{3}]$ for which Rolle's theorem holds.

Hence, the correct answer is option (B).

Question:96

For the function $\mathrm{f}(\mathrm{x})=\mathrm{x}+\frac{1}{\mathrm{x}}, \mathrm{x} \in[1,3]$ the value of c for mean value theorem is
A. 1

B. $\sqrt3$
C. 2
D. None of these

Answer: The
Mean Value Theorem states that, let f : [a, b] → R be a continuous function on [a, b] and differentiable on (a, b). Then there exists some c in (a, b) such that
$\\ f^{\prime}(c)=\frac{f(b)-f(a)}{b-a} \\ \\\text { We have, } f(x)=x+\frac{1}{x}$
Since f(x) is a polynomial function, it is continuous on [1,3] and differentiable on (1,3).
Now, as per the ean value Theorem, there exists at least one c ∈ (1,3), such that
$\\ f^{\prime}(c)=\frac{f(b)-f(a)}{b-a} \\ \Rightarrow 1-\frac{1}{c^{2}}=\frac{\left(3+\frac{1}{3}\right)-(1+1)}{3-1}\left[\because f^{\prime}(x)=1+\frac{1}{x^{2}}\right] \\ \Rightarrow \quad \frac{c^{2}-1}{c^{2}}=\frac{\left(\frac{10}{3}\right)-(2)}{2} \\ \Rightarrow \quad \frac{c^{2}-1}{c^{2}}=\frac{\left(\frac{10}{3}\right)-(2)}{2}=\frac{\frac{10-6}{3}}{2}=\frac{2}{3} \\ \Rightarrow 3\left(c^{2}-1\right)=2 c^{2} \\ \Rightarrow 3 c^{2}-2 c^{2}=3 \\ \Rightarrow c^{2}=3 \\ \Rightarrow c=\pm \sqrt{3} \\ \Rightarrow c=\sqrt{3} \in(1,3)$

Hence, the correct answer is option (B).

Question:97 Fill in the blanks in each of the
An example of a function that is continuous everywhere but fails to be differentiable exactly at two points is.

Answer:

Consider, f(x) = |x-1| + |x-2|
Let’s discuss the continuity of f(x).
We have, f(x) = |x-1| + |x-2|
$\begin{array}{l} f(x)=\left\{\begin{array}{l} -(x-1)-(x-2), \text { if } x<1 \\ (x-1)-(x-2), \text { if } 1 \leq x<2 \\ (x-1)+(x-2), \text { if } x \geq 2 \end{array}\right. \\ f(x)=\left\{\begin{array}{l} -2 x+3, \text { if } x<1 \\ 1, \text { if } 1 \leq x<2 \\ 2 x-3, \text { if } x \geq 2 \end{array}\right. \end{array}$
When x<1, we have f(x) = -2x+3, which is a polynomial function and the polynomial function is continuous everywhere.
When 1≤x<2, we have f(x) = 1, which is a constant function and the constant function is continuous everywhere.
When x≥2, we have f(x) = 2x-3, which is a polynomial function and the polynomial function is continuous everywhere.
Hence, f(x) = |x-1| + |x-2| is continuous everywhere.
Let’s discuss the differentiability of f(x) at x=1 and x=2.
We have
$\begin{array}{l} f(x)=\left\{\begin{array}{l} -(x-1)-(x-2), \text { if } x<1 \\ (x-1)-(x-2), \text { if } 1 \leq x<2 \\ (x-1)+(x-2), \text { if } x \geq 2 \end{array}\right. \\ f(x)=\left\{\begin{array}{l} -2 x+3, \text { if } x<1 \\ 1, \text { if } 1 \leq x<2 \\ 2 x-3, \text { if } x \geq 2 \end{array}\right. \end{array}$
$\\\qquad LHD =\lim _{x \rightarrow 1^{-}} \frac{f(x)-f(1)}{x-1} \\ =\lim _{h \rightarrow 0} \frac{f(1-h)-f(1)}{(1-h)-1} \\ =\lim _{h \rightarrow 0} \frac{[(-2(1-h)+3)-(-2+3)]}{(1-h)-1} \quad(\because f(x)=-2 x+3, \text { if } x<1)$
$\begin{aligned} &=\lim _{h \rightarrow 0} \frac{[-2+2 h+3-1]}{-h}\\ &\lim _{=h \rightarrow 0} \frac{[2 h]}{-h}=\lim _{h \rightarrow 0} 2=2\\ &RLD =\lim _{x \rightarrow 1^{+}} \frac{f(x)-f(1)}{x-1}\\ &=\lim _{x \rightarrow 1^{+}} \frac{1-1}{1-1}(\because f(x)=1, \text { if } 1 \leq x<2)\\ &=0\\ &\Rightarrow\left\lfloor f^{\prime}(1) \neq \operatorname{Rf}^{\prime}(1)\right.\\ &\Rightarrow f(x) \text { is not differentiable at } x=1 \text { . }\\ &L{ f^{\prime}(2)}=\lim _{x \rightarrow 2^{-}} \frac{f(x)-f(2)}{x-2} \end{aligned}$
$\begin{aligned} &=\lim _{x \rightarrow 2} \frac{1-1}{2-2}(: f(x)=1, \text { if } 1 \leq x<2 \text { and } f(2)=2 \times 2-3=1)\\ &=0\\ &{R f^{\prime}(2)}=\lim_{x \rightarrow 2^{+} }\frac{f(x)-f(2)}{x-2}\\ &=\lim _{h \rightarrow 0} \frac{f(1+h)-f(2)}{(1+h)-2}\\ &=\lim _{h \rightarrow 0} \frac{[(2(1+h)-3)-(2 \times 2-3)]}{(1+h)-2} \quad(\because f(x)=2 x-3, \text { if } x \geq 2)\\ &=\lim _{h \rightarrow 0} \frac{[2+2 h-3-1]}{h-1}\\ &=\lim _{h \rightarrow 0} \frac{[2 h-2]}{h-1}=\lim _{h \rightarrow 0} \frac{2(h-1)}{h-1}=2\\ &\Rightarrow \operatorname{Lf}^{\prime}(2) \neq \mathrm{Rf}^{\prime}(2) \end{aligned}$
⇒ f(x) is not differentiable at x=2.
Thus, f(x) = |x-1| + |X-2| is continuous everywhere but fails to be differentiable exactly at two points x=1 and x=2.

Question:98 Fill in the blanks in each of the
Derivative of $x^2$ w.r.t. $x^3$ is.

Answer:

$\begin{aligned} &\text { Let } u=x^{2} \text { and } v=x^{2}\\ &\Rightarrow \frac{d u}{d x}=2 x \text { and } \frac{d v}{d x}=3 x^{2}\\ &\therefore \frac{\mathrm{du}}{\mathrm{dv}}=\frac{\mathrm{du} / \mathrm{dx}}{\mathrm{dv} / \mathrm{dx}}=\frac{2 \mathrm{x}}{3 \mathrm{x}^{2}}\\ &\Rightarrow \frac{d u}{d v}=\frac{2}{3 x}\\ &\text { Hence, Derivative of }\\ &x^{2} \text { w.r.t. } x^{3} \text { is } \frac{2}{3 x} \text { . } \end{aligned}$

Question:99 Fill in the blanks in each of the
If $f(x) = |cos x|$, then ’$f'(\frac{\pi}{4})$ =.

Answer:

$\begin{aligned} &\text { We have, } f(x)=|\cos x|\\ &\text { Foc. } 0<x<\frac{\pi}{2}, \cos x>0\\ &\begin{array}{l} \therefore f(x)=\cos x \\ \Rightarrow f^{\prime}(x)=-\sin x \\ \Rightarrow f^{\prime}\left(\frac{\pi}{4}\right)=-\sin \frac{\pi}{4}=-\frac{1}{\sqrt{2}} \end{array}\\ &\text { Hence, } \\&f^{\prime}\left(\frac{\pi}{4}\right)=-\frac{1}{\sqrt{2}}\\ \end{aligned}$

Question:100 Fill in the blanks in each of the
If $f(x) = |cosx - sinx|$, then $f'\frac{\pi}{3}=$.

Answer:

We have, $f(x)=|\cos x-\sin x|$ Foc $\frac{\pi}{4}<x<\frac{\pi}{2}, \sin x>\cos x$

$\begin{aligned} & \therefore f(x)=\sin x-\cos x \\ & \Rightarrow f^{\prime}(x)=\cos x-(-\sin x)=\cos x+\sin x \\ & \Rightarrow f^{\prime}\left(\frac{\pi}{3}\right)=\frac{1}{2}+\frac{\sqrt{3}}{2}=\frac{1+\sqrt{3}}{2}\end{aligned}$

Hence, $f^{\prime}\left(\frac{\pi}{3}\right)=\frac{1+\sqrt{3}}{2}$

Question:101 Fill in the blanks in each of the
For the curve $\sqrt x + \sqrt y =1 , \frac{dy}{dx}\: \: at\: \: \left ( \frac{1}{4},\frac{1}{4} \right )$ is _______.

Answer:

We have, $\sqrt{x}+\sqrt{y}=1$

Differentiating with respect to x, we get,

$\begin{aligned} & \frac{1}{2 \sqrt{x}}+\frac{1}{2 \sqrt{y}} \frac{d y}{d x}=0 \\ & \Rightarrow \frac{1}{2 \sqrt{y}} \frac{d y}{d x}=-\frac{1}{2 \sqrt{x}} \\ & \Rightarrow \frac{d y}{d x}=-\frac{1}{2 \sqrt{x}} \times \frac{2 \sqrt{y}}{1} \\ & \Rightarrow \frac{d y}{d x}=-\frac{\sqrt{y}}{\sqrt{x}}\end{aligned}$

Now, $\left[\frac{d y}{d x}\right]_{\left(\frac{1}{4}, \frac{1}{4}\right)}=-\frac{\sqrt{\frac{1}{4}}}{\sqrt{\frac{1}{4}}}=-1$

Question:102 State True or False for the statements
Rolle’s theorem is applicable for the function f(x) = |x - 1| in [0, 2].

Answer:
As per Rolle’s Theorem, let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b), such that f(a) = f(b), where a and b are some real numbers. Then there exists some c in (a, b) such that f’(c) = 0.
We have f(x) = |x - 1| in [0, 2].
Since polynomial and modulus functions are continuous everywhere f(x) is continuous
Now, x-1=0
⇒ x=1
We need to check if f(x) is differentiable at x = 1 or not.
We have,
$\begin{aligned} &f(x)=\left\{\begin{array}{l} -(x-1), \text { if } x<1 \\ (x-1), \text { if } x>1 \end{array}\right.\\ &\lim _{x \rightarrow 1^{-}} \frac{f(x)-f(1)}{x-1}\\ &\lim _{h \rightarrow 0} \frac{f(1-h)-f(1)}{(1-h)-1}\\ &\lim _{h \rightarrow 0} \frac{[1-(1-h)-0]}{(1-h)-1} \quad(\because f(x)=1-x, \text { if } x<1)\\ &\lim _{n \rightarrow 0} \frac{[\mathrm{h}]}{-\mathrm{h}}\\ &\lim _{=h \rightarrow 0} \frac{[\mathrm{h}]}{-\mathrm{h}}=\lim _{\mathrm{h} \rightarrow 0}-1=-1\\ &\lim _{\mathrm{Rf}^{\prime}(1)}=\lim _{x \rightarrow 1^{+}} \frac{f(x)-f(1)}{x-1}\\ &\lim _{h \rightarrow 0} \frac{f(1+h)-f(1)}{(1+h)-1}\\ &\lim _{h \rightarrow 0} \frac{[(1+h)-1-0]}{(1+h)-1}(\because f(x)=x-1, \text { if } 1<x)\\ &\lim _{=h \rightarrow 0} \frac{[\mathrm{h}]}{\mathrm{h}}=\lim _{h \rightarrow 0} 1=1 \end{aligned}$
⇒ Lf’(1) ≠ Rf’(1)
⇒ f(x) is not differentiable at x=1.
Hence, Rolle’s theorem is not applicable to f(x) since it is not differentiable at x=1 ∈ (0,2).

Hence, the statement is False.

Question:103 State True or False for the statements
If f is continuous on its domain D, then |f| is also continuous on D.

Answer:
Given that, f is continuous on its domain D.
Let a be an arbitrary real number in D. Then f is continuous at a.
$\begin{aligned} & \lim _{x \rightarrow a} f(x)=f(a) \\ & \text { Now, } \\ & \lim _{x \rightarrow a}|f|(x)=\lim _{x \rightarrow a}|f(x)|[\because:|f|(x)=\mid f(x) L] \\ & \lim _{x \rightarrow a}|f|(x)=\left|\lim _{x \rightarrow a} f(x)\right| \lim _{x \rightarrow a}|f|(x)=|f(a)|=|f|(a)\end{aligned}$

If |f| is continuous at x=a.
Since a is an arbitrary point in D. Therefore |f| is continuous in D.

Hence, the statement is True.

Question:104 State True or False for the statements

The composition of two continuous functions is a continuous function.

Answer:
Let f be a function defined by f(x) = |1-x + |x||.
Let g(x) = 1-x + |x| and h(x) = |x| be two functions defined on R.
Then,
hog(x) = h(g(x))
⇒ hog(x) = h(1-x + |x|)
⇒ hog(x) = |(1-x + |x|)|
⇒ hog(x) = f(x) ∀ x ∈ R.
Since (1-x) is a polynomial function and |X| is a modulus function is continuous in R.
⇒ g(x) = 1-x + |x| is everywhere continuous.
⇒ h(x) = |x| is everywhere continuous.
Hence, f = hog is everywhere continuous.

Hence, the statement is True.

Question:105 State True or False for the statements
Trigonometric and inverse-trigonometric functions are differentiable in their respective domain.

Answer:

It is an obvious statement. Trigonometric and inverse-trigonometric functions are differentiable within their respective domains. Their derivatives are well-defined and commonly used in calculus.

Hence, the statement is True.

Question:106 State True or False for the statements
If f.g is continuous at x = a, then f and g are separately continuous at x = a.

Answer:
Let $f(x)=x$ and $g(x)=\frac{1}{x}$ $f(x) \cdot g(x)=x \cdot \frac{1}{x}=1$, which is a constant function and continuous everywhere.

But, $g(x)=\frac{1}{x}$ is discontinuous at $\mathrm{x}=0$.

Hence, the statement is False.

Students can use the NCERT Exemplar Class 12 Math Solutions Chapter 5 PDF download, prepared by experts, for a better understanding of concepts and topics of probability. The topics and subtopics are mentioned below.

Subtopics of NCERT Exemplar Class 12 Math Solutions Chapter 5 Continuity and Differentiability

The subtopics covered in this chapter are:

Introduction
Continuity
Algebra of continuous functions
Differentiability
Derivatives of composite functions
Derivatives of implicit functions
Derivatives of inverse trigonometric functions
Logarithmic and exponential functions
Logarithmic differentiation
Derivatives of functions in parametric forms
Second-order derivative
Mean value theorem

Importance of Solving NCERT Exemplar Class 12 Maths Solutions Chapter 5

The exemplar problems go beyond the basics, helping students grasp more advanced concepts with greater clarity.
Various types of questions, like MCQs, fill-in-the-blanks, true-false, short-answer type, and long-answer type, will enhance the logical and analytical skills of the students.
These NCERT exemplar exercises cover all the important topics and concepts so that students can be well-prepared for various exams.
By solving these NCERT exemplar problems, students will get to know about all the real-life applications of Continuity and Differentiability.

NCERT Exemplar Class 12 Maths Solutions Chapter

NCERT Exemplar for Class 12 Mathematics Chapter 1: Relations and Functions

NCERT Exemplar Class 12 Mathematics Chapter 2: Inverse Trigonometric Functions

NCERT Exemplar Class 12 Mathematics Chapter 3: Matrices

NCERT Exemplar Class 12 Mathematics Chapter 4: Determinants

NCERT Exemplar Class 12 Mathematics Chapter 5: Continuity and Differentiability

NCERT Exemplar Class 12 Mathematics Chapter 6: Applications of derivatives

NCERT Exemplar Class 12 Mathematics Chapter 7: Integrals

NCERT Exemplar Class 12 Mathematics Chapter 8: Applications of Integrals

NCERT Exemplar Class 12 Mathematics Chapter 9: Differential equations

NCERT Exemplar Class 12 Mathematics Chapter 10: Vector Algebra

NCERT Exemplar Class 12 Mathematics Chapter 11: Three-DimensionalGeometry

NCERT Exemplar Class 12 Mathematics Chapter 12: Linear Programming

NCERT Exemplar Class 12 Mathematics Chapter 13: Probability

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NCERT Exemplar Class 12 Solutions

NCERT Exemplar Class 12 Chemistry Solutions

NCERT Exemplar Class 12 Physics Solutions

NCERT Exemplar Class 12 Biology Solutions

Also, check NCERT Solutions for questions given in the book:

Chapter 1	Relations and Functions
Chapter 2	Inverse Trigonometric Functions
Chapter 3	Matrices
Chapter 4	Determinants
Chapter 5	Continuity and Differentiability
Chapter 6	Application of Derivatives
Chapter 7	Integrals
Chapter 8	Application of Integrals
Chapter 9	Differential Equations
Chapter 10	Vector Algebra
Chapter 11	Three-Dimensional Geometry
Chapter 12	Linear Programming
Chapter 13	Probability

Must Read NCERT Solution subject-wise

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Frequently Asked Questions (FAQs)

Q: What is the difference between continuity and differentiability?

Continuity and differentiability are closely related but distinct concepts in calculus. A function is continuous at a point if there is no break, jump, or hole, meaning the graph flows smoothly through that point. On the other hand, a function is differentiable at a point if it has a defined derivative there, implying the graph has a well-defined tangent and no sharp corners or cusps. All differentiable functions are continuous, but not all continuous functions are differentiable. For example, the absolute value function is continuous everywhere but not differentiable at x=0 due to a sharp corner.

Q: Can a function be differentiable but not continuous?

No, a function cannot be differentiable without being continuous. Differentiability implies continuity. If a function is differentiable at a point, it must also be continuous at that point. This is because finding a derivative involves calculating a limit, and for that limit to exist, the function must not break or jump at that point. However, the reverse is not always true, a function can be continuous but not differentiable, such as the absolute value function at x=0, which is continuous but has a sharp corner, making it non-differentiable at that poin

Q: What is the significance of Rolle’s theorem in differentiability?

Rolle's Theorem is significant in understanding the behavior of differentiable functions. It states that if a function is continuous on [a, b], differentiable on (a, b), and f(a)=f(b), then there exists at least one point c in (a, b) where f'(c)=0. This means the function has a horizontal tangent (slope zero) at some point between a and b. Rolle's Theorem helps in proving other important results like the Mean Value Theorem and is useful in analyzing turning points, ensuring the existence of critical points in various real-life applications involving motion or optimization.

Q: What is the concept of logarithmic differentiation?

Logarithmic differentiation is a technique used to differentiate complex functions, especially those involving products, quotients, or variables raised to variable powers. The process involves taking the natural logarithm (ln) of both sides of the function, simplifying using logarithmic identities, and then differentiating implicitly. This method is particularly useful when dealing with functions like y=x^x or y=(x²+1)^x, where standard rules are difficult to apply directly. Logarithmic differentiation simplifies calculations, making it easier to handle functions with multiple terms or exponents, and is a powerful tool in both calculus and applied mathematics.

Q: Why is every differentiable function always continuous, but not vice versa?

Every differentiable function is always continuous because differentiability requires the function's limit to exist and match the function's value at a point. In other words, for a function to be differentiable, it must first be smooth and unbroken-i.e., continuous. However, not all continuous functions are differentiable. A function can be continuous without having a defined derivative at some points, especially where there are sharp turns. For example, the absolute value function |x| is continuous everywhere but not differentiable at x=0, since it has a sharp corner at that point.

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12th Exam 2025Kitin tareekh ko hoga

Hello,

The date of 12 exam is depends on which board you belongs to . You should check the exact date of your exam by visiting the official website of your respective board.

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Nidhi Raghuwanshi

21 Sep'25

12th biology question paper 2023-2025

Hello,

Class 12 biology questions papers 2023-2025 are available on cbseacademic.nic.in , and other educational website. You can download PDFs of questions papers with solution for practice. For state boards, visit the official board site or trusted education portal.

Hope this information is useful to you.

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Nidhi Raghuwanshi

21 Sep'25

In March 2025, I passed the CBSE 12th board and gave KCET. I am taking a drop year. I will reappear in the CBSE board exam in the upcoming 2026 exam in February as a private candidate to increase my PCM aggregate, as KCET gives 50% weightage to PCM aggregate, and I will also give the KCET exam. So, to calculate rank, do they use the board marksheet from 2025 or 2026? Also for each subject will best out of two marks accross both the marksheets be considered? #kcet

Hello Pruthvi,

Taking a drop year to reappear for the Karnataka Common Entrance Test (KCET) is a well-defined process. As a repeater, you are fully eligible to take the exam again to improve your score and secure a better rank for admissions.

The main procedure involves submitting a new application for the KCET through the official Karnataka Examinations Authority (KEA) website when registrations open for the next academic session. You must pay the required application fee and complete all formalities just like any other candidate. A significant advantage for you is that you do not need to retake your 12th board exams. Your previously secured board marks in the qualifying subjects will be used again. Your new KCET rank will be calculated by combining these existing board marks with your new score from the KCET exam. Therefore, your entire focus during this year should be on preparing thoroughly for the KCET to achieve a higher score.

For more details about the KCET Exam preparation, CLICK HERE.

I hope this answer helps you. If you have more queries, feel free to share your questions with us, and we will be happy to assist you.

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Ark Pandey

20 Sep'25

I'm studying scince in karnatakastate board, can i take commerce in cbse for 12th next year?

Yes, you can switch from Science in Karnataka State Board to Commerce in CBSE for 12th. You will need a Transfer Certificate from your current school and meet the CBSE school’s admission requirements. Since you haven’t studied Commerce subjects like Accountancy, Economics, and Business Studies, you may need to catch up before or during 12th. Not all CBSE schools accept direct admission to 12th from another board, so some may ask you to join Class 11 first. Make sure to check the school’s rules and plan your subject preparation.

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Abhilasha Agrawal

18 Sep'25

12th class cbse boards most important que ans ans in hindi midum subject hindi impotent question jo exam main aaynge wo btai

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For the 12th CBSE Hindi Medium board exam, important questions usually come from core chapters like “Madhushala”, “Jhansi ki Rani”, and “Bharat ki Khoj”.
Questions often include essay writing, letter writing, and comprehension passages. Grammar topics like Tenses, Voice Change, and Direct-Indirect Speech are frequently asked.
Students should practice poetry questions on themes and meanings. Important questions also cover summary writing and translation from Hindi to English or vice versa.
Previous years’ question papers help identify commonly asked questions.
Focus on writing practice to improve handwriting and presentation. Time management during exams is key to answering all questions effectively.

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JEE Test Series

Option 1) $0.34\; J$	Option 2) $0.16\; J$
Option 3) $1.00\; J$	Option 4) $0.67\; J$

Option 1) $2,000 \; J - 5,000\; J$	Option 2) $200 \, \, J - 500 \, \, J$
Option 3) $2\times 10^{5}J-3\times 10^{5}J$	Option 4) $20,000 \, \, J - 50,000 \, \, J$

Option 1) $K/2\,$	Option 2) $\; K\;$
Option 3) $zero\;$	Option 4) $K/4$

Option 1) $11.2\, L\, H_{2(g)}$ at STP is produced for every mole $HCL_{(aq)}$ consumed	Option 2) $6L\, HCl_{(aq)}$ is consumed for ever $3L\, H_{2(g)}$ produced
Option 3) $33.6 L\, H_{2(g)}$ is produced regardless of temperature and pressure for every mole Al that reacts	Option 4) $67.2\, L\, H_{2(g)}$ at STP is produced for every mole Al that reacts .

Option 1) 0.02	Option 2) 3.125 × 10^-2
Option 3) 1.25 × 10^-2	Option 4) 2.5 × 10^-2

Option 1) decrease twice	Option 2) increase two fold
Option 3) remain unchanged	Option 4) be a function of the molecular mass of the substance.

Option 1) Molality	Option 2) Weight fraction of solute
Option 3) Fraction of solute present in water	Option 4) Mole fraction.

Option 1) twice that in 60 g carbon	Option 2) 6.023 × 10²²
Option 3) half that in 8 g He	Option 4) 558.5 × 6.023 × 10²³

Option 1) less than 3	Option 2) more than 3 but less than 6
Option 3) more than 6 but less than 9	Option 4) more than 9

NCERT Exemplar Class 12 Maths Solutions Chapter 5 Continuity and Differentiability

Subtopics of NCERT Exemplar Class 12 Math Solutions Chapter 5 Continuity and Differentiability

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