Class 12 Maths Chapter 5 Continuity and Differentiability – Exercise 5.2 NCERT Solutions

Introduction

Exercise 5.2 focuses on differentiability of functions. Students learn how to check differentiability at a point, apply derivative rules, and understand the relationship between continuity and differentiability. This exercise is essential for mastering calculus and preparing for integration techniques.

Formulas Used

1. Differentiability at a Point: A function $f(x)$ is differentiable at $x=a$ if:

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

2. Relation Between Continuity and Differentiability: Differentiability ⇒ Continuity, but continuity does not always ⇒ differentiability.

3. Basic Derivatives:

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

$$\frac{d}{dx}(e^x)=e^x,\frac{d}{dx}(\ln x)=\frac{1}{x}$$

Students Frequently Make Mistakes

• Forgetting that differentiability implies continuity but not vice versa.

• Confusing left‑hand derivative with right‑hand derivative.

• Errors in applying derivative formulas.

• Ignoring domain restrictions in logarithmic and trigonometric functions.

• Skipping verification of continuity before checking differentiability.

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

Q1. Check differentiability of $f(x)=x^2$ at $x=2$.

$$f^{\mathrm{\prime }}(x)=2x,f^{\mathrm{\prime }}(2)=4$$

Differentiable at $x=2$.

Q2. Check differentiability of $f(x)=\mathrm{\mid }x\mathrm{\mid }$ at $x=0$. Left derivative: ${\lim }_{h\to 0^{-}}\frac{\mathrm{\mid }h\mathrm{\mid }-0}{h}={\lim }_{h\to 0^{-}}\frac{-h}{h}=-1$. Right derivative: ${\lim }_{h\to 0^{+}}\frac{\mathrm{\mid }h\mathrm{\mid }-0}{h}={\lim }_{h\to 0^{+}}\frac{h}{h}=1$. Not equal ⇒ not differentiable at 0.

Q3. Differentiate $f(x)=\sin x$.

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

Q4. Differentiate $f(x)=\cos x$.

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

Q5. Differentiate $f(x)=e^x$.

$$f^{\mathrm{\prime }}(x)=e^x$$

Q6. Differentiate $f(x)=\ln x$.

$$f^{\mathrm{\prime }}(x)=\frac{1}{x},x>0$$

Q7. Differentiate $f(x)=x^3$.

$$f^{\mathrm{\prime }}(x)=3x^2$$

Q8. Differentiate $f(x)=\tan x$.

$$f^{\mathrm{\prime }}(x)={\sec }^{2}x,x\mathrm{\ne }\frac{\pi }{2}+n\pi$$

Q9. Differentiate $f(x)=x^n$.

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

Q10. Check differentiability of $f(x)=\sqrt{x}$ at $x=4$.

$$f^{\mathrm{\prime }}(x)=\frac{1}{2\sqrt{x}},f^{\mathrm{\prime }}(4)=\frac{1}{4}$$

Differentiable at $x=4$.

FAQs (10)

FAQ1. What is differentiability at a point? Existence of derivative at that point.

FAQ2. Does differentiability imply continuity? Yes.

FAQ3. Does continuity imply differentiability? Not always. Example: $\mathrm{\mid }x\mathrm{\mid }$ at 0.

FAQ4. What is derivative of $x^n$? $n{x}^{n-1}$.

FAQ5. What is derivative of $\sin x$? $\cos x$.

FAQ6. What is derivative of $\cos x$? $-\sin x$.

FAQ7. What is derivative of $e^x$? $e^x$.

FAQ8. What is derivative of $\ln x$? $\frac{1}{x},x>0$.

FAQ9. What is derivative of $\tan x$? ${\sec }^{2}x$.

FAQ10. Why is Exercise 5.2 important? It builds foundation for calculus and advanced differentiation.

Conclusion

Exercise 5.2 has 10 solved questions and 10 FAQs that strengthen your understanding of differentiability and its relation to continuity. This builds the foundation for advanced calculus in Class 12 Maths.

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