# NCERT Solutions for Class 11 Maths Maths Chapter 13

## Limits and Derivatives Class 11

### Exercise 13.1 : Solutions of Questions on Page Number : 301

Q1 :

Evaluate the Given limit:  Q2 :

Evaluate the Given limit:  Q3 :

Evaluate the Given limit: Q4 :

Evaluate the Given limit: Q5 :

Evaluate the Given limit: Q6 :

Evaluate the Given limit: Q7 :

Evaluate the Given limit: Q8 :

Evaluate the Given limit: Q9 :

Evaluate the Given limit: Q10 :

Evaluate the Given limit: Q11 :

Evaluate the Given limit: Q12 :

Evaluate the Given limit: Q13 :

Evaluate the Given limit: Q14 :

Evaluate the Given limit: Q15 :

Evaluate the Given limit: Q16 :

Evaluate the given limit: Q17 :

Evaluate the Given limit: Q18 :

Evaluate the Given limit: Q19 :

Evaluate the Given limit: Q20 :

Evaluate the Given limit: Q21 :

Evaluate the Given limit: Q22 : Q23 :

Find f(x) and f(x), where f(x) = Q24 :

Find f(x), where f(x) = Q25 :

Evaluate f(x), where f(x) = Q26 :

Find f(x), where f(x) = Q27 :

Find f(x), where f(x) = Q28 :

Suppose f(x) = and if f(x) = f(1) what are possible values of a and b?

Q29 :

Let be fixed real numbers and define a function What is f(x)? For some compute f(x).

Q30 :

If f(x) = .

For what value (s) of a does f(x) exists?

Q31 :

If the function f(x) satisfies , evaluate .

Q32 :

If . For what integers m and n does and exist?

### Exercise 13.2 : Solutions of Questions on Page Number : 312

Q1 :

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

Q2 :

Find the derivative of 99x at x = 100.

Q3 :

Find the derivative of x at x = 1.

Q4 :

Find the derivative of the following functions from first principle.

(i) x3 â€“ 27 (ii) (x â€“ 1) (x â€“ 2)

(ii) (iv) Q5 :

For the function Prove that Q6 :

Find the derivative of for some fixed real number a.

Q7 :

For some constants a and b, find the derivative of

(i) (x â€“ a) (x â€“ b) (ii) (ax2 + b)2 (iii) Q8 :

Find the derivative of for some constant a.

Q9 :

Find the derivative of

(i) (ii) (5x3 + 3x â€“ 1) (x â€“ 1)

(iii) xâ€“3 (5 + 3x) (iv) x5 (3 â€“ 6xâ€“9)

(v) xâ€“4 (3 â€“ 4xâ€“5) (vi) Q10 :

Find the derivative of cos x from first principle.

Q11 :

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

### Exercise Miscellaneous : Solutions of Questions on Page Number : 317

Q1 :

Find the derivative of the following functions from first principle:

(i) â€“x (ii) (â€“x)â€“1 (iii) sin (x + 1)

(iv) Q2 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x + a)

Q3 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): Q4 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax + b) (cx + d)2

Q5 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): Q6 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): Q7 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): Q8 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): Q9 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): Q10 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): Q11 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): Q12 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax + b)n

Q13 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax + b)n (cx + d)m

Q14 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): sin (x + a)

Q15 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): cosec x cot x

Q16 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): Q17 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): Q18 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): Q19 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): sinn x

Q20 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): Q21 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): Q22 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): x4 (5 sin x - 3 cos x)

Q23 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x2 + 1) cos x

Q24 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax2 + sin x) (p + q cos x)

Q25 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): Q26 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): Q27 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): Q28 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): Q29 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (x + sec x) (x - tan x)

Q30 :

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 