NCERT Solutions for Miscellaneous Exercise Chapter 12 Class 11 - Limits and Derivatives

Question 1:(i) Find the derivative of the following functions from first principle: -x

Answer:

Given.

f(x)=-x

Now, As we know, The derivative of any function at x is

$f^{\prime }(x)=\underset{h\to 0}{lim}\frac{f(x+h)-f(x)}{h}$

$f^{\prime }(x)=\underset{h\to 0}{lim}\frac{-(x+h)-(-x)}{h}$

$f^{\prime }(x)=\underset{h\to 0}{lim}\frac{-h}{h}$

$f^{\prime }(x)=-1$

Question 1:(ii) Find the derivative of the following functions from first principle: $(-x{)}^{-1}$

Answer:

Given.

f(x)= $(-x{)}^{-1}$

Now, As we know, The derivative of any function at x is

$f^{\prime }(x)=\underset{h\to 0}{lim}\frac{f(x+h)-f(x)}{h}$

$f^{\prime }(x)=\underset{h\to 0}{lim}\frac{-(x+h{)}^{-1}-(-x{)}^{-1}}{h}$

$f^{\prime }(x)=\underset{h\to 0}{lim}\frac{-\frac{1}{x+h}+\frac{1}{x}}{h}$

$f^{\prime }(x)=\underset{h\to 0}{lim}\frac{\frac{-x+x+h}{x(x+h)}}{h}$

$f^{\prime }(x)=\underset{h\to 0}{lim}\frac{\frac{h}{(x+h)(x)}}{h}$

$f^{\prime }(x)=\underset{h\to 0}{lim}\frac{1}{x(x+h)}$

$f^{\prime }(x)=\frac{1}{x^2}$

Question 1:(iii) Find the derivative of the following functions from first principle: $\sin (x+1)$

Answer:

Given.

$f(x)=\sin (x+1)$

Now, As we know, The derivative of any function at x is

$f^{\prime }(x)=\underset{h\to 0}{lim}\frac{f(x+h)-f(x)}{h}$

$f^{\prime }(x)=\underset{h\to 0}{lim}\frac{\sin (x+h+1)-\sin (x+1)}{h}$

$f^{\prime }(x)=\underset{h\to 0}{lim}\frac{2\cos (\frac{x+h+1+x+1}{2})\sin (\frac{x+h+1-x-1}{2})}{h}$

$f^{\prime }(x)=\underset{h\to 0}{lim}\frac{2\cos (\frac{2x+h+2}{2})\sin (\frac{h}{2})}{h}$

$f^{\prime }(x)=\underset{h\to 0}{lim}\frac{\cos (\frac{2x+h+2}{2})\sin (\frac{h}{2})}{\frac{h}{2}}$

$f^{\prime }(x)=\cos (\frac{2x+0+2}{2})\times 1$

$f^{\prime }(x)=\cos (x+1)$

Question 1:(iv) Find the derivative of the following functions from first principle: $\cos (x-\pi /8)$

Answer:

Given.

$f(x)=\cos (x-\pi /8)$

Now, As we know, The derivative of any function at x is

$f^{\prime }(x)=\underset{h\to 0}{lim}\frac{f(x+h)-f(x)}{h}$

$f^{\prime }(x)=\underset{h\to 0}{lim}\frac{\cos (x+h-\pi /8)-cos(x-\pi /8)}{h}$

$f^{\prime }(x)=\underset{h\to 0}{lim}\frac{-2\sin \left(\frac{x+h-\pi /8+x-\pi /8}{2}\right)\sin \left(\frac{x+h-\pi /8-x+\pi /8}{2}\right)}{h}$

$f^{\prime }(x)=\underset{h\to 0}{lim}\frac{-2\sin \left(\frac{2x+h-\pi /4}{2}\right)\sin \left(\frac{h}{2}\right)}{h}$

$f^{\prime }(x)=\underset{h\to 0}{lim}\frac{-\sin \left(\frac{2x+h-\pi /4}{2}\right)\sin \left(\frac{h}{2}\right)}{\frac{h}{2}}$

$f^{\prime }(x)=\sin \left(\frac{2x+0-\pi /4}{2}\right)\times 1$

$f^{\prime }(x)=-\sin (x-\pi /8)$

Question 2: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x+a )

Answer:

Given

f(x)= x + a

As we know, the property,

$f^{\prime }(x^n)=n{x}^{n-1}$

applying that property we get

$f^{\prime }(x)=1+0$

$f^{\prime }(x)=1$

Question 3: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $(px+q)(\frac{r}{x}+s)$

Answer:

Given

$f(x)=(px+q)(\frac{r}{x}+s)$

$f(x)=pr+psx+\frac{qr}{x}+qs$

As we know, the property,

$f^{\prime }(x^n)=n{x}^{n-1}$

applying that property we get

$f^{\prime }(x)=0+ps+\frac{-qr}{x^2}+0$

$f^{\prime }(x)=ps+q\left(\frac{-r}{x^2}\right)$

$f^{\prime }(x)=ps-\left(\frac{qr}{x^2}\right)$

Question 4: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $(ax+b)(cx+d{)}^2$

Answer:

Given,

$f(x)=(ax+b)(cx+d{)}^2$

$f(x)=(ax+b)(c^2{x}^2+2cdx+d^2)$

$f(x)=a{c}^2{x}^3+2acd{x}^2+a{d}^{2}x+b{c}^2{x}^2+2bcdx+b{d}^2$

Now,

As we know, the property,

$f^{\prime }(x^n)=n{x}^{n-1}$

and the property

$\frac{d(y_1+y_2)}{dx}=\frac{d{y}_1}{dx}+\frac{d{y}_2}{dx}$

applying that property we get

$f^{\prime }(x)=3a{c}^2{x}^2+4acdx+a{d}^2+2b{c}^{2}x+2bcd+0$

$f^{\prime }(x)=3a{c}^2{x}^2+4acdx+a{d}^2+2b{c}^{2}x+2bcd$

$f^{\prime }(x)=3a{c}^2{x}^2+(4acd+2b{c}^2)x+a{d}^2+2bcd$

Question 5: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $ax+b/cx+d$

Answer:

Given,

$f(x)=\frac{ax+b}{cx+d}$

Now, As we know the derivative of any function

$\frac{d(\frac{y_1}{y_2})}{dx}=\frac{y_{2}d(\frac{d{y}_1}{dx})-y_1(\frac{d{y}_2}{dx})}{y_2^2}$

Hence, The derivative of f(x) is

$\frac{d(\frac{ax+b}{cx+d})}{dx}=\frac{(cx+d)d(\frac{d(ax+b)}{dx})-(ax+b)(\frac{d(cx+d)}{dx})}{(cx+d{)}^2}$

$\frac{d(\frac{ax+b}{cx+d})}{dx}=\frac{(cx+d)a-(ax+b)c}{(cx+d{)}^2}$

$\frac{d(\frac{ax+b}{cx+d})}{dx}=\frac{acx+ad-acx-bc}{(cx+d{)}^2}$

$\frac{d(\frac{ax+b}{cx+d})}{dx}=\frac{ad-bc}{(cx+d{)}^2}$

Hence Derivative of the function is

$\frac{ad-bc}{(cx+d{)}^2}$ .

Question 6: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

$\frac{1+\frac{1}{x}}{1-\frac{1}{x}}$

Answer:

Given,

$f(x)=\frac{1+\frac{1}{x}}{1-\frac{1}{x}}$

Also can be written as

$f(x)=\frac{x+1}{x-1}$

Now, As we know the derivative of any function

$\frac{d(\frac{y_1}{y_2})}{dx}=\frac{y_{2}d(\frac{d{y}_1}{dx})-y_1(\frac{d{y}_2}{dx})}{y_2^2}$

Hence, The derivative of f(x) is

$\frac{d(\frac{x+1}{x-1})}{dx}=\frac{(x-1)d(\frac{d(x+1)}{dx})-(x+1)(\frac{d(x-1)}{dx})}{(x-1{)}^2}$

$\frac{d(\frac{x+1}{x-1})}{dx}=\frac{(x-1)1-(x+1)1}{(x-1{)}^2}$

$\frac{d(\frac{x+1}{x-1})}{dx}=\frac{x-1-x-1}{(x-1{)}^2}$

$\frac{d(\frac{x+1}{x-1})}{dx}=\frac{-2}{(x-1{)}^2}$

Hence Derivative of the function is

$\frac{-2}{(x-1{)}^2}$

Question 7: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

$\frac{1}{a{x}^2+bx+c}$

Answer:

Given,

$f(x)=\frac{1}{a{x}^2+bx+c}$

Now, As we know the derivative of any such function is given by

$\frac{d(\frac{y_1}{y_2})}{dx}=\frac{y_{2}d(\frac{d{y}_1}{dx})-y_1(\frac{d{y}_2}{dx})}{y_2^2}$

Hence, The derivative of f(x) is

$\frac{d(\frac{1}{a{x}^2+bx+c})}{dx}=\frac{(a{x}^2+bx+c)d(\frac{d(1)}{dx})-1(\frac{d(a{x}^2+bx+c)}{dx})}{(a{x}^2+bx+c{)}^2}$

$\frac{d(\frac{1}{a{x}^2+bx+c})}{dx}=\frac{0-(2ax+b)}{(a{x}^2+bx+c{)}^2}$

$\frac{d(\frac{1}{a{x}^2+bx+c})}{dx}=-\frac{(2ax+b)}{(a{x}^2+bx+c{)}^2}$

Question 8: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

$\frac{ax+b}{p{x}^2+qx+r}$

Answer:

Given,

$f(x)=\frac{ax+b}{p{x}^2+qx+r}$

Now, As we know the derivative of any function

$\frac{d(\frac{y_1}{y_2})}{dx}=\frac{y_{2}d(\frac{d{y}_1}{dx})-y_1(\frac{d{y}_2}{dx})}{y_2^2}$

Hence, The derivative of f(x) is

$\frac{d(\frac{ax+b}{p{x}^2+qx+r})}{dx}=\frac{(p{x}^2+qx+r)d(\frac{d(ax+b)}{dx})-(ax+b)(\frac{d(p{x}^2+qx+r)}{dx})}{(p{x}^2+qx+r{)}^2}$

$\frac{d(\frac{ax+b}{p{x}^2+qx+r})}{dx}=\frac{(p{x}^2+qx+r)a-(ax+b)(2px+q)}{(p{x}^2+qx+r{)}^2}$

$\frac{d(\frac{ax+b}{p{x}^2+qx+r})}{dx}=\frac{ap{x}^2+aqx+ar-2ap{x}^2-aqx-2bpx-bq}{(p{x}^2+qx+r{)}^2}$

$\frac{d(\frac{ax+b}{p{x}^2+qx+r})}{dx}=\frac{-ap{x}^2+ar-2bpx-bq}{(p{x}^2+qx+r{)}^2}$

Question 9: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

$\frac{p{x}^2+qx+r}{ax+b}$

Answer:

Given,

$f(x)=\frac{p{x}^2+qx+r}{ax+b}$

Now, As we know the derivative of any function

$\frac{d(\frac{y_1}{y_2})}{dx}=\frac{y_{2}d(\frac{d{y}_1}{dx})-y_1(\frac{d{y}_2}{dx})}{y_2^2}$

Hence, The derivative of f(x) is

$\frac{d(\frac{p{x}^2+qx+r}{ax+b})}{dx}=\frac{(ax+b)d(\frac{d(p{x}^2+qx+r)}{dx})-(p{x}^2+qx+r)(\frac{d(ax+b)}{dx})}{(ax+b{)}^2}$

$\frac{d(\frac{p{x}^2+qx+r}{ax+b})}{dx}=\frac{(ax+b)(2px+q)-(p{x}^2+qx+r)(a)}{(ax+b{)}^2}$

$\frac{d(\frac{p{x}^2+qx+r}{ax+b})}{dx}=\frac{2ap{x}^2+aqx+2bpx+bq-ap{x}^2-aqx-ar}{(ax+b{)}^2}$

$\frac{d(\frac{p{x}^2+qx+r}{ax+b})}{dx}=\frac{ap{x}^2+2bpx+bq-ar}{(ax+b{)}^2}$

Question 10: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

$\frac{a}{x^4}-\frac{b}{x^2}+\cos x$

Answer:

Given

$f(x)=\frac{a}{x^4}-\frac{b}{x^2}+\cos x$

As we know, the property,

$f^{\prime }(x^n)=n{x}^{n-1}$

and the property

$\frac{d(y_1+y_2)}{dx}=\frac{d{y}_1}{dx}+\frac{d{y}_2}{dx}$

applying that property we get

$f^{\prime }(x)=\frac{-4a}{x^5}-(\frac{-2b}{x^3})+(-\sin x)$

$f^{\prime }(x)=\frac{-4a}{x^5}+\frac{2b}{x^3}-\sin x$

Question 11: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $4\sqrt{x}-2$

Answer:

Given

$f(x)=4\sqrt{x}-2$

It can also be written as

$f(x)=4x^{\frac{1}{2}}-2$

Now,

As we know, the property,

$f^{\prime }(x^n)=n{x}^{n-1}$

and the property

$\frac{d(y_1+y_2)}{dx}=\frac{d{y}_1}{dx}+\frac{d{y}_2}{dx}$

applying that property we get

$f^{\prime }(x)=4(\frac{1}{2})x^{-\frac{1}{2}}-0$

$f^{\prime }(x)=2x^{-\frac{1}{2}}$

$f^{\prime }(x)=\frac{2}{\sqrt{x}}$

Question 12: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

$(ax+b{)}^n$

Answer:

Given

$f(x)=(ax+b{)}^n$

Now, As we know the chain rule of derivative,

$[f(g(x)){]}^{\prime }=f^{\prime }(g(x))\times g^{\prime }(x)$

And, the property,

$f^{\prime }(x^n)=n{x}^{n-1}$

Also the property

$\frac{d(y_1+y_2)}{dx}=\frac{d{y}_1}{dx}+\frac{d{y}_2}{dx}$

applying those properties we get,

$f^{\prime }(x)=n(ax+b{)}^{n-1}\times a$

$f^{\prime }(x)=an(ax+b{)}^{n-1}$

Question 13: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $(ax+b{)}^n(cx+d{)}^m$

Answer:

Given

$f(x)=(ax+b{)}^n(cx+d{)}^m$

Now, As we know the chain rule of derivative,

$[f(g(x)){]}^{\prime }=f^{\prime }(g(x))\times g^{\prime }(x)$

And the Multiplication property of derivative,

$\frac{d(y_1{y}_2)}{dx}=y_1\frac{d{y}_2}{dx}+y_2\frac{d{y}_1}{dx}$

And, the property,

$f^{\prime }(x^n)=n{x}^{n-1}$

Also the property

$\frac{d(y_1+y_2)}{dx}=\frac{d{y}_1}{dx}+\frac{d{y}_2}{dx}$

Applying those properties we get,

$f^{\prime }(x)=(ax+b{)}^n(m(cx+d{)}^{m-1})+(cx+d)(n(ax+b{)}^{n-1})$

$f^{\prime }(x)=m(ax+b{)}^n(cx+d{)}^{m-1}+n(cx+d)(ax+b{)}^{n-1}$

Question 14: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $\sin (x+a)$

Answer:

Given,

$f(x)=\sin (x+a)$

Now, As we know the chain rule of derivative,

$[f(g(x)){]}^{\prime }=f^{\prime }(g(x))\times g^{\prime }(x)$

Applying this property we get,

$f^{\prime }(x)=\cos (x+a)\times 1$

$f^{\prime }(x)=\cos (x+a)$

Question 15: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $\csc x\cot x$

Answer:

Given,

$f(x)=\csc x\cot x$

the Multiplication property of derivative,

$\frac{d(y_1{y}_2)}{dx}=y_1\frac{d{y}_2}{dx}+y_2\frac{d{y}_1}{dx}$

Applying the property

$\frac{d(\csc x)(\cot x))}{dx}=\csc x\frac{d\cot x}{dx}+\cot x\frac{d\csc x}{dx}$

$\frac{d(\csc x)(\cot x))}{dx}=\csc x(-{\csc }^{2}x)+\cot x(-\csc x\cot x)$

$\frac{d(\csc x)(\cot x))}{dx}=-{\csc }^{3}x-{\cot }^{2}x\csc x$

Hence derivative of the function is $-{\csc }^{3}x-{\cot }^{2}x\csc x$ .

Question 16: Find the derivative of the following functions (it is to be understood that a, b, c, d,p, q, r and s are fixed non-zero constants and m and n are integers):

$\frac{\cos x}{1+\sin x}$

Answer:

Given,

$f(x)=\frac{\cos x}{1+\sin x}$

Now, As we know the derivative of any function

$\frac{d(\frac{\cos x}{1+\sin x})}{dx}=\frac{(1+\sin x)d(\frac{d\cos x}{dx})-\cos x(\frac{d(1+\sin x)}{dx})}{(1+\sin x{)}^2}$

Hence, The derivative of f(x) is

$\frac{d(\frac{\cos x}{1+\sin x})}{dx}=\frac{(1+\sin x)(-\sin x)-\cos x(\cos x)}{(1+\sin x{)}^2}$

$\frac{d(\frac{\cos x}{1+\sin x})}{dx}=\frac{-\sin x-{\sin }^{2}x-{\cos }^{2}x}{(1+\sin x{)}^2}$

$\frac{d(\frac{\cos x}{1+\sin x})}{dx}=\frac{-\sin x-1}{(1+\sin x{)}^2}$

$\frac{d(\frac{\cos x}{1+\sin x})}{dx}=-\frac{1}{(1+\sin x)}$

Question 17: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $\frac{\sin x+\cos x}{\sin x-\cos x}$

Answer:

Given

$f(x)=\frac{\sin x+\cos x}{\sin x-\cos x}$

Also can be written as

$f(x)=\frac{\tan x+1}{\tan x-1}$

which further can be written as

$f(x)=-\frac{\tan x+tan(\pi /4)}{1-\tan (\pi /4)\tan x}$

$f(x)=-\tan (x-\pi /4)$

Now,

$f^{\prime }(x)=-{\sec }^2(x-\pi /4)$

$f^{\prime }(x)=-\frac{1}{{\cos }^2(x-\pi /4)}$

Question 18: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

$\frac{\sec x-1}{\sec x+1}$

Answer:

Given,

$f(x)=\frac{\sec x-1}{\sec x+1}$

which also can be written as

$f(x)=\frac{1-\cos x}{1+\cos x}$

Now,

As we know the derivative of such function

$\frac{d(\frac{y_1}{y_2})}{dx}=\frac{y_{2}d(\frac{d{y}_1}{dx})-y_1(\frac{d{y}_2}{dx})}{y_2^2}$

So, The derivative of the function is,

$\frac{d(\frac{1-\cos x}{1+\cos x})}{dx}=\frac{(1+\cos x)d(\frac{d(1-\cos x)}{dx})-(1-\cos x)(\frac{d(1+\cos x)}{dx})}{(1+\cos x{)}^2}$

$\frac{d(\frac{1-\cos x}{1+\cos x})}{dx}=\frac{(1+\cos x)(-(-\sin x))-(1-\cos x)(-\sin x)}{(1+\cos x{)}^2}$

$\frac{d(\frac{1-\cos x}{1+\cos x})}{dx}=\frac{\sin x+\sin x\cos x+\sin x-\cos x\sin x}{(1+\cos x{)}^2}$

$\frac{d(\frac{1-\cos x}{1+\cos x})}{dx}=\frac{2\sin x}{(1+\cos x{)}^2}$

Which can also be written as

$\frac{d(\frac{1-\cos x}{1+\cos x})}{dx}=\frac{2\sec x\tan x}{(1+\sec x{)}^2}$ .

Question 19: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): ${\sin }^{n}x$

Answer:

Given,

$f(x)={\sin }^{n}x$

Now, As we know the chain rule of derivative,

$[f(g(x)){]}^{\prime }=f^{\prime }(g(x))\times g^{\prime }(x)$

And, the property,

$f^{\prime }(x^n)=n{x}^{n-1}$

Applying those properties, we get

$f^{\prime }(x)=n{\sin }^{n-1}x\cos x$

Hence Derivative of the given function is $n{\sin }^{n-1}x\cos x$

Question 20: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

$\frac{a+b\sin x}{c+d\cos x}$

Answer:

Given Function

$f(x)=\frac{a+b\sin x}{c+d\cos x}$

Now, As we know the derivative of any function of this type is:

$\frac{d(\frac{y_1}{y_2})}{dx}=\frac{y_{2}d(\frac{d{y}_1}{dx})-y_1(\frac{d{y}_2}{dx})}{y_2^2}$

Hence derivative of the given function will be:

$\frac{d(\frac{a+b\sin x}{c+d\cos x})}{dx}=\frac{(c+d\cos x)(\frac{d(a+b\sin x)}{dx})-(a+b\sin x)(\frac{d(c+d\cos xx)}{dx})}{(c+d\cos x{)}^2}$

$\frac{d(\frac{a+b\sin x}{c+d\cos x})}{dx}=\frac{(c+d\cos x)(b\cos x)-(a+b\sin x)(d(-\sin x))}{(c+d\cos x{)}^2}$

$\frac{d(\frac{a+b\sin x}{c+d\cos x})}{dx}=\frac{cb\cos x+bd{\cos }^{2}x+ad\sin x+bd{\sin }^{2}x}{(c+d\cos x{)}^2}$

$\frac{d(\frac{a+b\sin x}{c+d\cos x})}{dx}=\frac{cb\cos x+ad\sin x+bd({\sin }^{2}x+{\cos }^{2}x)}{(c+d\cos x{)}^2}$

$\frac{d(\frac{a+b\sin x}{c+d\cos x})}{dx}=\frac{cb\cos x+ad\sin x+bd}{(c+d\cos x{)}^2}$

Question 21: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):

$\frac{\sin (x+a)}{\cos x}$

Answer:

Given,

$f(x)=\frac{\sin (x+a)}{\cos x}$

Now, As we know the derivative of any function

$\frac{d(\frac{y_1}{y_2})}{dx}=\frac{y_{2}d(\frac{d{y}_1}{dx})-y_1(\frac{d{y}_2}{dx})}{y_2^2}$