Class 11 Maths Chapter 12 Limits and Derivatives – Exercise 12.2 NCERT Solutions

Introduction

Exercise 12.2 introduces the concept of derivatives in calculus. You will learn how to calculate derivatives of functions using the first principle and standard rules. This exercise builds the foundation for differentiation techniques and applications in higher mathematics.

Key Concepts

1. Derivative Definition (First Principle):

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

2. Standard Derivatives:

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

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

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

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

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

3. Derivative of Constant:

$$\frac{d}{dx}(c)=0$$

NCERT Questions with Solutions (10)

Q1. Find derivative of $x^2$ using first principle.

$$f(x)=x^2,f^{\mathrm{\prime }}(x)=\underset{h\to 0}{\lim }\frac{(x+h{)}^2-x^2}{h}=\underset{h\to 0}{\lim }\frac{2xh+h^2}{h}=2x$$

Q2. Find derivative of $x^3$ using first principle.

$$f(x)=x^3,f^{\mathrm{\prime }}(x)=\underset{h\to 0}{\lim }\frac{(x+h{)}^3-x^3}{h}=\underset{h\to 0}{\lim }\frac{3x^{2}h+3x{h}^2+h^3}{h}=3x^2$$

Q3. Find derivative of $\sin x$ using first principle.

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

Q4. Find derivative of $\cos x$ using first principle.

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

Q5. Find derivative of $e^x$.

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

Q6. Find derivative of $\ln x$.

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

Q7. Find derivative of constant 7.

$$\frac{d}{dx}(7)=0$$

Q8. Find derivative of $x^5$.

$$\frac{d}{dx}(x^5)=5x^4$$

Q9. Find derivative of $\sin 2x$.

$$\frac{d}{dx}(\sin 2x)=2\cos 2x$$

Q10. Find derivative of $\cos 3x$.

$$\frac{d}{dx}(\cos 3x)=-3\sin 3x$$

FAQs (10)

1. What is derivative? Rate of change of function with respect to variable.

2. What is first principle of derivative? Limit definition: ${\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$.

3. What is derivative of constant? Zero.

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

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

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

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

8. What is derivative of $\ln x$? $\frac{1}{x}$.

9. Can derivative be negative? Yes, depending on slope of function.

10. Why is this exercise important? It builds foundation for differentiation techniques and calculus applications.

Conclusion

Exercise 12.2 has 10 solved questions and 10 FAQs that strengthen your understanding of derivatives using first principles and standard rules. This builds the foundation for advanced differentiation in Class 11 Maths.

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New Syllabus-Class 11 Maths Chapter 12 Limits and Derivatives – Exercise 12.2 NCERT