**NCERT Solutions for Class 11 Maths Chapter 12 ****Limits and Derivatives**** Ex 12.2**

**Ex 12.2 Class 11 Maths Question 1.**

### Find the derivative of x^{2} – 2 at x = 10.

**Solution:**

Let f(x) = x^{2} – 2 …(i)

Differentiating (i) with respect to x, we get

f'(x) = 2x

At x = 10, f'(10) = 2(10) = 20.

**Ex 12.2 Class 11 Maths Question 2.**

### Find the derivative of x at x = 1.

**Solution:**

Let f(x) = x …(i)

Differentiating (i) with respect to x, we get

f'(x) = 1

At x = 1, f'(1) = 1.

**Ex 12.2 Class 11 Maths Question 3.**

### Find the derivative of 99x at x = 100.

**Solution:**

Let f(x) = 99x …(i)

Differentiating (i) with respect to x, we get

f'(x) = 99

At x = 100, f'(100) = 99.

**Ex 12.2 Class 11 Maths Question 4.**

### Find the derivative of the following functions from first principle.

**(i)** x^{3} – 27

**(ii)** (x – 1) (x – 2)

**(iii)** 1/x^{2}

**(iv)** (x+1)/(x−1)

**Solution:**

**Ex 12.2 Class 11 Maths Question 5.**

### For the function

Prove that f'(1) = 100f'(0).

**Solution:**

Given,

**Ex 12.2 Class 11 Maths Question 6.**

### Find the derivative of x^{n} + ax^{n-1} + a^{2}x^{n-2 }+ …. + a^{n-1}x + a^{n} for some fixed real number a.

**Solution:**

Let f(x) = x^{n} + ax^{n-1} + a^{2}x^{n-2 }+ …. + a^{n-1}x + a^{n } …(i)

Differentiating (i) with respect to x, we get

f'(x) = nx^{n-1} + (n – 1)ax^{n-2} + …… + a^{n-1}

**Ex 12.2 Class 11 Maths Question 7.**

### For some constants a and b, find the derivative of

**(i)** (x – a) (x – b)

**(ii)** (ax^{2} + b)^{2}

**(iii)** (x−a)/(x−b)

**Solution:****(i)** Let f(x) = (x – a) (x – b) ….(1)

Differentiating (1) with respect to x, we get

f'(x) = (x – a) (x – b)’ + (x – a)’ (x – b)

⇒ f'(x) = (x – a) + (x – b) = 2x – a – b

**Ex 12.2 Class 11 Maths Question 8.**

### Find the derivative (x^{n}−a^{n})/(x−a) for some constant a.

**Solution:**

Let f(x) = (x^{n}−a^{n})/(x−a), for some constant a^{ } ….(i)

Differentiating (i) with respect to x, we get

**Ex 12.2 Class 11 Maths Question 9.**

### Find the derivative of

**Solution:****(i)** Let f(x) = 2x – 3/4 …(1)

Differentiating (1) with respect to x, we get

f'(x) = 2 – 0 ⇒ f'(x) = 2.**(ii)** Let f(x) = (5x^{3} + 3x – 1)(x – 1)

**Ex 12.2 Class 11 Maths Question 10.**

### Find the derivative of cos x from first principle.

**Solution:**

Let f(x) = cos x

**Ex 12.2 Class 11 Maths Question 11.**

### Find the derivative of the following functions:

**(i)** sin x cos x

**(ii)** sec x

**(iii)** 5 sec x + 4 cos x

**(iv)** cosec x

**(v)** 3 cot x + 5 cosec x

**(vi)** 5 sin x – 6 cos x + 7

**(vii)** 2 tan x – 7 sec x

**Solution:****(i)** Let f(x) = sin x cos x … (1)

Differentiating (1) with respect to x, we get

**Related Links:**

**NCERT Solutions for Maths Class 9**