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
Question:1. Differentiate the following w.r.t. 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:
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:
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:
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,
The main topics covered in Chapter 5 of continuity and differentiability, exercises 5.4 are:
Also Read,
Below are some useful links for subject-wise NCERT solutions for class 12.
Here are some links to subject-wise solutions for the NCERT exemplar class 12.
Frequently Asked Questions (FAQs)
The derivative of e^x is e^x.
The derivative of log(x) is 1/x.
The derivative of sin(x) is cos(x).
The derivative of cos(x) is -sin(x).
The derivative of sin(x^2) is 2x cos(x^2).
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.
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.
The derivative of tan (x) is sec^2(x).
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