NCERT Solutions for Exercise 5.4 Class 12 Maths Chapter 5 - Continuity and Differentiability

NCERT Solutions for Exercise 5.4 Class 12 Maths Chapter 5 - Continuity and Differentiability

Komal MiglaniUpdated on 23 Apr 2025, 11:09 PM IST

If continuity is like playing a song without any pauses, then differentiability means that there are no high pitches and sudden jumps in tone, just silky smooth transitions. Finding the derivatives of exponential and logarithmic functions is an important aspect of calculus. In exercise 5.4 of the chapter Continuity and Differentiability, we explore different methods to easily differentiate functions involving exponential and logarithmic functions. This article on the NCERT Solutions for exercise 5.4, class 12 maths chapter 5 - Continuity and Differentiability, offers an easy-to-understand and step-by-step solution for the problems present in the exercise, so that students will get a better understanding of the methods and logic. For syllabus, notes, and PDF, refer to this link: NCERT

Class 12 Maths Chapter 5 Exercise 5.4 Solutions: Download PDF

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Continuity and Differentiability Exercise: 5.4

Question:1. Differentiate the following w.r.t. x:

$\frac{e ^x }{\sin x }$

Answer:

Given function is
$f(x)=\frac{e ^x }{\sin x }$
We differentiate with the help of Quotient rule
$f^{'}(x) = \frac{\frac{d(e^x)}{dx} \cdot \sin x - e^x \cdot \frac{d(\sin x)}{dx}}{\sin^2 x} = \frac{e^x \cdot \sin x - e^x \cdot \cos x}{\sin^2 x} = \frac{e^x(\sin x - \cos x)}{\sin^2 x}$
Therefore, the answer is $\frac{e^x(\sin x-\cos x)}{\sin^2x}$

Question:2. Differentiate the following w.r.t. x:

$e^{\sin^{-1}x}$

Answer:

Given function is
$f(x)=e ^{\sin ^{-1}x}$
Let $g(x)={\sin ^{-1}x}$
Then,
$f(x)=e^{g(x)}$
Now, differentiation w.r.t. x
$f^{'}(x)=g^{'}(x).e^{g(x)}$ -(i)
$g(x) = \sin^{-1}x \Rightarrow g^{'}(x ) = \frac{1}{\sqrt{1-x^2}}$
Put this value in our equation (i)
$f^{'}(x) = \frac{1}{\sqrt{1-x^2}}.e^{\sin^{-1}x} = \frac{e^{\sin^{-1}x}}{\sqrt{1-x^2}}$

Question:3. Differentiate the following w.r.t. x:

$e^{x^3}$

Answer:

Given function is
$f(x)=e ^{x^3}$
Let $g(x)=x^3$
Then,
$f(x)=e^{g(x)}$
Now, differentiation w.r.t. x
$f^{'}(x)=g^{'}(x).e^{g(x)}$ -(i)

$g(x) = x^3 \Rightarrow g'(x) = 3x^2$

Put this value in our equation (i)
$f^{'}(x) =3x^2.e^{x^3}$
Therefore, the answer is $3x^2.e^{x^3}$

Question:4. Differentiate the following w.r.t. x:

$\sin ( \tan ^ { -1} e ^{-x })$

Answer:

Given function is
$f(x)=\sin ( \tan ^ { -1} e ^{-x })$
Let's take $g(x ) = \tan^{-1}e^{-x}$
Now, our function reduces to
$f(x) = \sin(g(x))$
Now,
$f^{'}(x) = g^{'}(x)\cos(g(x))$ -(i)
And
$g(x)=\tan^{-1}e^{-x}\\\Rightarrow g^{'}(x) = \frac{d(\tan^{-1}e^{-x})}{dx}.\frac{d(e^{-x})}{dx}= \frac{1}{1+(e^{-x})^2}.-e^{-x} = \frac{-e^{-x}}{1+e^{-2x}}$
Put this value in our equation (i)
$f^{'}(x) =\frac{-e^{-x}}{1+e^{-2x}}\cos(\tan^{-1}e^{-x})$
Therefore, the answer is $\frac{-e^{-x}}{1+e^{-2x}}\cos(\tan^{-1}e^{-x})$

Question:5. Differentiate the following w.r.t. x:

$\log (\cos e ^x )$

Answer:

Given function is
$f(x)=\log (\cos e ^x )$
Let's take $g(x ) = \cos e^{x}$
Now, our function reduces to
$f(x) = \log(g(x))$
Now,
$f^{'}(x) = g^{'}(x).\frac{1}{g(x)}$ -(i)
And
$g(x)=\cos e^{x}\\\Rightarrow g^{'}(x) = \frac{d(\cos e^{x})}{dx}.\frac{d(e^{x})}{dx}= (-\sin e^x).e^{x} = -e^x.\sin e^x$
Put this value in our equation (i)
$f^{'}(x) =-e^x.\sin e^x.\frac{1}{\cos e^x} = -e^x.\tan e^x \ \ \ \ \ (\because \frac{\sin x}{\cos x}=\tan x)$
Therefore, the answer is $-e^x.\tan e^x,\ \ \ e^x\neq (2n+1)\frac{\pi}{2},\ \ n\in N$

Question:6. Differentiate the following w.r.t. x:

$e ^x + e ^{x^2} + .....e ^{x^5}$

Answer:

Given function is
$f(x)= e ^x + e ^{x^2} + .....e ^{x^5}$
Now, differentiation w.r.t. x is
$f^{'}(x)= \frac{d(e^x)}{dx}.\frac{d(x)}{dx}+\frac{d(e^{x^2})}{dx}.\frac{d(x^2)}{dx}+\frac{d(e^{x^3})}{dx}.\frac{d(x^3)}{dx}+\frac{d(e^{x^4})}{dx}.\frac{d(x^4)}{dx}+\frac{d(e^{x^5})}{dx}.\frac{d(x^5)}{dx}$
$=e^x.1+e^{x^2}.2x+e^{x^3}.3x^2+e^{x^4}.4x^3+e^{x^5}.5x^4$
$=e^x+2xe^{x^2}+3x^2e^{x^3}+4x^3e^{x^4}+5x^4e^{x^5}$
Therefore, answer is $e^x+2xe^{x^2}+3x^2e^{x^3}+4x^3e^{x^4}+5x^4e^{x^5}$

Question:7. Differentiate the following w.r.t. x:

$\sqrt { e ^{ \sqrt x }} , x > 0$

Answer:

Given function is
$f(x)=\sqrt { e ^{ \sqrt x }}$
Lets take $g(x ) = \sqrt x$
Now, our function reduces to
$f(x) = \sqrt {e^{g(x)}}$
Now,
$f^{'}(x) = g^{'}(x) \cdot \frac{1}{2\sqrt{e^{g(x)}}} \cdot \frac{d\left(e^{g(x)}\right)}{dx} = g^{'}(x) \cdot \frac{1}{2\sqrt{e^{g(x)}}} \cdot e^{g(x)} = \frac{g^{'}(x) \cdot e^{g(x)}}{2\sqrt{e^{g(x)}}} = \frac{g^{'}(x) \cdot e^{\sqrt{x}}}{2\sqrt{e^{\sqrt{x}}}} \text{ -(i)}$
And
$g(x)=\sqrt x\\\Rightarrow g^{'}(x) = \frac{(\sqrt x)}{dx}=\frac{1}{2\sqrt x}$
Put this value in our equation (i)
$f^{'}(x) = \frac{e^{\sqrt{x}}}{2\sqrt{x} \cdot 2\sqrt{e^{\sqrt{x}}}} = \frac{e^{\sqrt{x}}}{4\sqrt{x e^{\sqrt{x}}}}$
Therefore, the answer is $\frac{e^{\sqrt x}}{4\sqrt{xe^{\sqrt x}}}.\ \ x>0$

Question:8 Differentiate the following w.r.t. x: $\log ( \log x ) , x > 1$

Answer:

Given function is
$f(x)=\log ( \log x )$
Lets take $g(x ) = \log x$
Now, our function reduces to
$f(x) = \log(g(x))$
Now,
$f^{'}(x) = g^{'}(x).\frac{1}{g(x)}$ -(i)
And
$g(x)=\log x\\\Rightarrow g^{'}(x) = \frac{1}{x}$
Put this value in our equation (i)
$f^{'}(x) =\frac{1}{x}.\frac{1}{\log x} = \frac{1}{x\log x}$
Therefore, the answer is $\frac{1}{x\log x}, \ \ x>1$

Question:9. Differentiate the following w.r.t. x:

$\frac{\cos x }{\log x} , x > 0$

Answer:

Given function is
$f(x)=\frac{\cos x }{\log x}$
We differentiate with the help of Quotient rule
$f^{'}(x)=\frac{\frac{d(\cos x)}{dx}.\log x-\cos x.\frac{(\log x)}{dx} }{(\log x)^2 }$
$=\frac{(-\sin x).\log x-\cos x.\frac{1}{x} }{(\log x)^2 } = \frac{-(x\sin x\log x+\cos x)}{x(\log x)^2}$
Therefore, the answer is $\frac{-(x\sin x\log x+\cos x)}{x(\log x)^2}$

Question:10. Differentiate the following w.r.t. x:

$\cos ( log x + e ^x ) , x > 0$

Answer:

Given function is
$f(x)=\cos ( log x + e ^x )$
Lets take $g(x) = ( log x + e ^x )$
Then , our function reduces to
$f(x) = \cos (g(x))$
Now, differentiation w.r.t. x is
$f'(x) = g'(x)(-\sin(g(x))) \tag{i}$
And
$g(x) = ( log x + e ^x )$
$g^{'}(x)= \frac{d(\log x)}{dx}+\frac{d(e^x)}{dx} = \frac{1}{x}+e^x$
Put this value in our equation (i)
$f^{'}(x) = -\left ( \frac{1}{x}+e^x \right )\sin (\log x+e^x)$
Therefore, the answer is $-\left ( \frac{1}{x}+e^x \right )\sin (\log x+e^x), x>0$


Also Read,

Topics covered in Chapter 5, Continuity and Differentiability: Exercise 5.4

The main topics covered in Chapter 5 of continuity and differentiability, exercises 5.4 are:

  • Derivatives of Exponential functions: Differentiation of exponential functions is very basic in calculus. In particular derivative of $e^x$ is very simple.
    $\frac{d}{dx}(e^x)=e^x$
    Also, for the general exponent function, the derivative is as follows:
    $\frac{d}{dx}(a^x)=a^x\ln a$
  • Derivatives of Logarithmic functions: Logarithmic functions help in simplifying many complex calculations. The basic differentiation rule of a logarithmic function is:
    $\frac{d}{dx}(\ln x)=\frac{1}{x}$

Also Read,

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NCERT Exemplar Solutions Subject Wise

Here are some links to subject-wise solutions for the NCERT exemplar class 12.

Frequently Asked Questions (FAQs)

Q: What is the derivative of e^x ?
A:

The derivative of e^x is e^x.

Q: What is the derivative of log(x) ?
A:

The derivative of log(x) is 1/x.

Q: what is the derivative of sin(x) ?
A:

The derivative of sin(x) is cos(x).

Q: what is the derivative of cos(x) ?
A:

The derivative of cos(x) is -sin(x).

Q: What is the derivative of sin(x^2) ?
A:

The derivative of sin(x^2) is 2x cos(x^2).

Q: Which is the best book for Maths CBSE Class 12 ?
A:

NCERT textbook is the best book for CBSE board exams which you should follow. You don't need any additional books for the CBSE board exams.

Q: How may questions are there in the exercise 5.4 Class 12 Maths ?
A:

A total of 10 questions are there in exercise 5.4 CBSE Class 12 Maths. For more questions students can refer to the NCERT exemplar questions.

Q: Find the derivative of tan (x) ?
A:

The derivative of tan (x) is sec^2(x).

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