# Chapter 13 Limits and Derivatives Ex – 13.2

Question 1.

Find the derivative of x2 – 2 at x = 10.

Solution:

let f(x) = x2 – 2

Differentiating (i) with respect to x, we get

f'(x) = 2x

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

Question 2.

Find the derivative of 99x at x = 10.

Solution:

let f(x) = 99x

Differentiating (i) with respect to x, we get

f'(x) = 90

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

Question 3.

Find the derivative of x at x = 10.

Solution:

let f(x) = x

Differentiating (i) with respect to x, we get

f'(x) = 1

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

Question 4.

Find the derivative of the following functions from first principle.

(i) x3 – 27

(ii) (x – 1)(x – 2)

(iii) $\frac { 1 }{ { x }^{ 2 } }$

(iv) $\frac { x+1 }{ x-1 }$

Solution:

Question 5.

For the function

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

Solution:

We have

Question 6.

Find the derivative of xn + axn-1 + a2xn-2+

…. + an-1x + an for some fixed real number a.

Solution:

Let f(x) = xn + axn-1 + a2xn-2+

…. + an-1x + an

Differentiating (i) with respect to x, we get

f'(x) = nxn-1 + (n – 1)axn-2 + …… + an-1

Question 7.

For some constants a and b, find the derivative of

(i) (x – a)(x – b)

(ii) (ax2 + b)2

(iii) $\frac { x-a }{ x-b }$

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

Question 8.

Find the derivative $\frac { { x }^{ n }-{ a }^{ n } }{ x-a }$ for some constant a.

Solution:

Let f(x) = $\frac { { x }^{ n }-{ a }^{ n } }{ x-a }$ ….(i), where a is a constant.

Differentiating (i) with respect to x, we get

Question 9.

Find the derivative of

Solution:

(i) Let f(x) = $2x-\frac { 3 }{ 4 }$ …(1)
Differentiating (i) with respect to x, we get
f'(x) = 2·1 – 0 ⇒ f'(x) = 2.
(ii) Let f(x) = 5x3 + 3x – 1)(x – 1)

Question 10.

Find the derivative of cos x from the first principle.

Solution:

Let f(x) = cos x

Question 11.

Find the derivative of the following functions:

(i) sin x cos x
(ii) secx
(iii) 5 secx + 4 cosx
(iv) cosecx
(v) 3 cotx + 5 cosecx
(vi) 5sinx – 6 cosx + 7
(vii) 2 tanx – 7 secx.

Solution:

(i) Let f(x) = sin x cos x … (1)
Differentiating (1) with respect to x, we get