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

Introduction

Exercise 5.4 focuses on exponential and logarithmic functions and their derivatives. Students learn how to differentiate functions involving $e^x$, $\ln x$, and composite forms like $a^x$ or ${\log }_{a}x$. This exercise is essential for calculus, integration, and applications in growth/decay models.

Formulas Used

1. Exponential Function:

$$\frac{d}{dx}(e^x)=e^x$$

$$\frac{d}{dx}(a^x)=a^x\ln a,a>0,a\mathrm{\ne }1$$

2. Logarithmic Function:

$$\frac{d}{dx}(\ln x)=\frac{1}{x},x>0$$

$$\frac{d}{dx}({\log }_{a}x)=\frac{1}{x\ln a},a>0,a\mathrm{\ne }1$$

3. Chain Rule:

$$\frac{d}{dx}[f(g(x))]=f^{\mathrm{\prime }}(g(x))\cdot g^{\mathrm{\prime }}(x)$$

Students Frequently Make Mistakes

• Forgetting domain restrictions ($x>0$ for logarithmic functions).

• Missing factor $\ln a$ in derivative of $a^x$.

• Confusing natural log ($\ln x$) with log base 10 ($\log x$).

• Errors in applying chain rule.

• Skipping simplification of final answer.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. Differentiate $y=e^x$.

$$\frac{dy}{dx}=e^x$$

Q2. Differentiate $y=a^x$.

$$\frac{dy}{dx}=a^x\ln a$$

Q3. Differentiate $y=\ln x$.

$$\frac{dy}{dx}=\frac{1}{x},x>0$$

Q4. Differentiate $y={\log }_{a}x$.

$$\frac{dy}{dx}=\frac{1}{x\ln a}$$

Q5. Differentiate $y=e^{2x}$.

$$\frac{dy}{dx}=2e^{2x}$$

Q6. Differentiate $y=\ln (2x+1)$.

$$\frac{dy}{dx}=\frac{1}{2x+1}\cdot 2=\frac{2}{2x+1}$$

Q7. Differentiate $y=a^{x^2}$.

$$\frac{dy}{dx}=a^{x^2}\cdot 2x\ln a$$

Q8. Differentiate $y=\ln (\sin x)$.

$$\frac{dy}{dx}=\frac{1}{\sin x}\cdot \cos x=\cot x$$

Q9. Differentiate $y=\ln (\tan x)$.

$$\frac{dy}{dx}=\frac{1}{\tan x}\cdot {\sec }^{2}x=\frac{{\sec }^{2}x}{\tan x}$$

Q10. Differentiate $y=e^{\ln x}$. Since $e^{\ln x}=x$,

$$\frac{dy}{dx}=1$$

FAQs (10)

FAQ1. What is derivative of $e^x$? Itself, $e^x$.

FAQ2. What is derivative of $a^x$? $a^x\ln a$.

FAQ3. What is derivative of $\ln x$? $\frac{1}{x},x>0$.

FAQ4. What is derivative of ${\log }_{a}x$? $\frac{1}{x\ln a}$.

FAQ5. Why domain restriction for $\ln x$? Because log defined only for $x>0$.

FAQ6. What is derivative of $e^{kx}$? $k{e}^{kx}$.

FAQ7. What is derivative of $\ln (f(x))$? $\frac{f^{\mathrm{\prime }}(x)}{f(x)}$.

FAQ8. What is derivative of $a^{f(x)}$? $a^{f(x)}\cdot f^{\mathrm{\prime }}(x)\ln a$.

FAQ9. Why use chain rule? To differentiate composite functions.

FAQ10. Why is Exercise 5.4 important? It builds foundation for calculus involving exponential and logarithmic functions.

Conclusion

Exercise 5.4 has 10 solved questions and 10 FAQs that strengthen your understanding of exponential and logarithmic derivatives. This builds the foundation for advanced calculus in Class 12 Maths.

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