NCERT Solutions for Class 11 Maths Chapter 13- Limits and Derivatives

Exercise 13.1 page no: 301

1. Evaluate the Given limit:

Solution:

Given

Substituting x = 3, we get

= 3 + 3

= 6

2. Evaluate the Given limit:

Solution:

Given limit:

Substituting x = π, we get

= (π – 22 / 7)

3. Evaluate the Given limit:

Solution:

Given limit:

Substituting r = 1, we get

= π(1)²

= π

4. Evaluate the Given limit:

Solution:

Given limit:

Substituting x = 4, we get

= [4(4) + 3] / (4 – 2)

= (16 + 3) / 2

= 19 / 2

5. Evaluate the Given limit:

Solution:

Given limit:

Substituting x = -1, we get

= [(-1)¹⁰ + (-1)⁵ + 1] / (-1 – 1)

= (1 – 1 + 1) / – 2

= – 1 / 2

6. Evaluate the Given limit:

Solution:

Given limit:

= [(0 + 1)⁵ – 1] / 0

=0

Since, this limit is undefined

Substitute x + 1 = y, then x = y – 1

7. Evaluate the Given limit:

Solution:

8. Evaluate the Given limit:

Solution:

9. Evaluate the Given limit:

Solution:

= [a (0) + b] / c (0) + 1

= b / 1

= b

10. Evaluate the Given limit: 

Solution:

11. Evaluate the Given limit:

Solution:

Given limit:

Substituting x = 1

= [a (1)² + b (1) + c] / [c (1)² + b (1) + a]

= (a + b + c) / (a + b + c)

Given

= 1

12. Evaluate the Given limit:

Solution:

By substituting x = – 2, we get

13. Evaluate the Given limit:

Solution:

Given 

14. Evaluate the given limit: 

Solution:

15. Evaluate the given limit:

Solution:

16. Evaluate the given limit:

Solution:

17. Evaluate the given limit:

Solution:

18. Evaluate the given limit:

Solution:

19. Evaluate the given limit:

Solution:

20. Evaluate the given limit:

Solution:

21. Evaluate the given limit:

Solution:

22. Evaluate the given limit:

Solution:

23.

Solution:

24. Find
, where

Solution:

25. Evaluate
, where f(x) =

Solution:

26. Find
, where f (x) =

Solution:

27. Find
, where

Solution:

28. Suppose
and if
what are possible values of a and b

Solution:

29. Let a₁, a_2,………a_n be fixed real numbers and define a function

f (x) = (x – a₁) (x – a₂) ……. (x – a_n).

What is
For some a ≠ a₁, a₂, ……. a_n, compute

Solution:

30. If  For what value (s) of a does exists?

Solution:

31. If the function f(x) satisfies , evaluate

Solution:

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

Solution:

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Exercise 13.2 page no: 312

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

Solution:

Let f (x) = x² – 2

From first principle

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

Solution:

Let f (x) = x

Then,

3. Find the derivative of 99x at x = l00.

Solution:

Let f (x) = 99x,

From first principle

= 99

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

(i) x³ – 27

(ii) (x – 1) (x – 2)

(iii) 1 / x²

(iv) x + 1 / x – 1

Solution:

(i) Let f (x) = x³ – 27

From first principle

(ii) Let f (x) = (x – 1) (x – 2)

From first principle

(iii) Let f (x) = 1 / x²

From first principle, we get

(iv) Let f (x) = x + 1 / x – 1

From first principle, we get

5. For the function  .Prove that f’ (1) =100 f’ (0).

Solution:

6. Find the derivative of  for some fixed real number a.

Solution:

7. For some constants a and b, find the derivative of
(i) (x − a) (x − b)

(ii) (ax² + b)²

(iii) x – a / x – b

Solution:

(i) (x – a) (x – b)

(ii) (ax² + b)²

= 4ax (ax² + b)

(iii) x – a / x – b

8. Find the derivative of  for some constant a.

Solution:

9. Find the derivative of

(i) 2x – 3 / 4

(ii) (5x³ + 3x – 1) (x – 1)

(iii) x^-3 (5 + 3x)

(iv) x⁵ (3 – 6x^-9)

(v) x^-4 (3 – 4x^-5)

(vi) (2 / x + 1) – x² / 3x – 1

Solution:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

10. Find the derivative of cos x from first principle

Solution:

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) 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

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Miscellaneous exercise page no: 317

1. Find the derivative of the following functions from first principle:

(i) –x 

(ii) (–x)^–1 

(iii) sin (x + 1)

(iv) 

Solution:

(ii) (-x)^-1

= 1 / x²

(iii) sin (x + 1)

(iv) 

We get,

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):

2. (x + a)

Solution:

3. (px + q) (r / x + s)

Solution:

4. (ax + b) (cx + d)²

Solution:

5. (ax + b) / (cx + d)

Solution:

6. (1 + 1 / x) / (1 – 1 / x)

Solution:

7. 1 / (ax² + bx + c)

Solution:

8. (ax + b) / px² + qx + r

Solution:

9. (px² + qx + r) / ax + b

Solution:

10. (a / x⁴) – (b / x²) + cox x

Solution:

11. 

Solution:

12. (ax + b)ⁿ

Solution:

13. (ax + b)ⁿ (cx + d)^m

Solution:

14. sin (x + a)

Solution:

15. cosec x cot x

Solution:

So, we get

16. 

Solution:

17.

Solution:

18.

Solution:

19. sinⁿ x

Solution:

20. 

Solution:

21.

Solution:

22. x⁴ (5 sin x – 3 cos x)

Solution:

23. (x² + 1) cos x

Solution:

24. (ax² + sin x) (p + q cos x)

Solution:

25. 

Solution:

26. 

Solution:

27. 

Solution:

28. 

Solution:

29. (x + sec x) (x – tan x)

Solution:

30. 

Solution: