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

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

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Question 1:

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

Answer:

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.

Answer:

Let f(x) = 99x. Accordingly,

Thus, the derivative of 99x at x = 100 is 99.

Question 3:

Find the derivative of x at x = 1.

Answer:

Let f(x) = x. Accordingly,

Thus, the derivative of x at x = 1 is 1.

Question 4:

Find the derivative of the following functions from first principle.

(i) x³ – 27 (ii) (x – 1) (x – 2)

(ii)  (iv) 

Answer:

(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 

Answer:

The given function is

Thus,

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Question 6:

Find the derivative offor some fixed real number a.

Answer:

Let 

Question 7:

For some constants a and b, find the derivative of

(i) (x – a) (x – b) (ii) (ax² + b)² (iii) 

Answer:

(i) Let f (x) = (x – a) (x – b)

(ii) Let 

(iii) 

By quotient rule,

Question 8:

Find the derivative offor some constant a.

Answer:

By quotient rule,

Question 9:

Find the derivative of

(i)  (ii) (5x³ + 3x – 1) (x – 1)

(iii) x^–3 (5 + 3x) (iv) x⁵ (3 – 6x^–9)

(v) x^–4 (3 – 4x^–5) (vi) 

Answer:

(i) Let

(ii) Let f (x) = (5x³ + 3x – 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 f (x) = x^–4 (3 – 4x^–5)

By Leibnitz product rule,

(vi) Let f (x) = 

By quotient rule,

Question 10:

Find the derivative of cos x from first principle.

Answer:

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

Answer:

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

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

(iii) Let f (x) = 5 sec x + 4 cos x. Accordingly, from the first principle,

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

(v) Let f (x) = 3cot x + 5cosec x. Accordingly, from the first principle,

From (1), (2), and (3), we obtain

(vi) Let f (x) = 5sin x – 6cos x + 7. Accordingly, from the first principle,

(vii) Let f (x) = 2 tan x – 7 sec x. Accordingly, from the first principle,