## NCERT Solutions for Class 11 Maths Chapter 13 – Limits and Derivatives Ex 13.2

#### Question 1:

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

Let f(x) = x2 – 2. Accordingly,
Thus, the derivative of x2 – 2 at x = 10 is 20.

#### Question 2:

Find the derivative of 99x at x = 100.

Let f(x) = 99x. Accordingly,
Thus, the derivative of 99x at x = 100 is 99.

#### Question 3:

Find the derivative of at = 1.

Let f(x) = x. Accordingly,
Thus, the derivative of at = 1 is 1.

#### Question 4:

Find the derivative of the following functions from first principle.
(i) x3 – 27 (ii) (x – 1) (– 2)
(ii)  (iv)

(i) Let f(x) = x3 – 27. Accordingly, from the first principle,
(ii) Let f(x) = (x – 1) (x – 2). Accordingly, from the first principle,
(iii) Let. Accordingly, from the first principle,
(iv) Let. Accordingly, from the first principle,

#### Question 5:

For the function
Prove that

The given function is
Thus,

#### Question 6:

Find the derivative offor some fixed real number a.

Let

#### Question 7:

For some constants a and b, find the derivative of
(i) (– a) (x – b) (ii) (ax2 + b)2 (iii)

(i) Let f (x) = (– a) (x – b)
(ii) Let
(iii)
By quotient rule,

#### Question 8:

Find the derivative offor some constant a.

By quotient rule,

#### Question 9:

Find the derivative of
(i)  (ii) (5x3 + 3– 1) (x – 1)
(iii) x–3 (5 + 3x) (iv) x5 (3 – 6x–9)
(v) x–4 (3 – 4x–5) (vi)

(i) Let
(ii) Let f (x) = (5x3 + 3– 1) (x – 1)
By Leibnitz product rule,
(iii) Let f (x) = x– 3 (5 + 3x)
By Leibnitz product rule,
(iv) Let f (x) = x5 (3 – 6x–9)
By Leibnitz product rule,
(v) Let (x) = x–4 (3 – 4x–5)
By Leibnitz product rule,
(vi) Let (x) =
By quotient rule,

#### Question 10:

Find the derivative of cos x from first principle.

Let f (x) = cos x. Accordingly, from the first principle,

#### 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) 3cot x + 5cosec x
(vi) 5sin x – 6cos x + 7 (vii) 2tan x – 7sec x