# RD SHARMA Solutions for Class 12-science Maths Chapter 11 - Differentiation

Page / Exercise

## Chapter 11 - Differentiation Exercise Ex. 11VSAQ

Question 30

If f(x) = x + 7 and g(x) = x - 7, x R, then find

Solution 30

Given: f(x) = x + 7 and g(x) = x - 7

Now, (fog)(x) = f(g(x)) = f(x - 7) = x - 7 + 7 = x

Therefore, (fog)(x) = x

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Write the derivative of sin x with respect to cos x.

Solution 28

Question 29

If  then find

Solution 29

Given:

## Chapter 11 - Differentiation Exercise MCQ

Question 33

If  then  is equal to

1.
2.
3.
4.
Solution 33

Given:

Differentiating w.r.t x, we get

Question 1

If f(x) = logx2 (log x), then f' (x) at x = e is

a. 0

b. 1

c. 1/e

d. 1/2 e

Solution 1

Correct option: (d)

Question 2

The differential coefficient of f (log x) with respect to x, where f (x) = log x is

Solution 2

Correct option: (c)

Question 3

Solution 3

Correct option: (a)

Question 4

Differential coefficient of sec (tan-1 x) is

Solution 4

Correct option: (d)

Question 5

Solution 5

Correct option: (d)

Question 6

Solution 6

Correct option: (a)

Question 7

Solution 7

Correct option: (d)

Question 8

Given f(x) = 4x8, then

Solution 8

Correct option: (c)

Question 9

Solution 9

Correct option: (d)

Question 10

Solution 10

Correct option:(a)

Question 11

Solution 11

Correct option: (a)

Question 12

Solution 12

Correct option: (c)

Question 13

Solution 13

Correct option: (d)

Question 14

1. 1/2
2. x

1. 1
Solution 14

Correct option: (d)

Question 15

Solution 15

Correct option: (b)

Question 16

Solution 16

Correct option: (a)

Question 17

Solution 17

Correct option: (d)

Question 18

Solution 18

Correct option: (a)

Question 19

Solution 19

Correct option: (b)

Question 20

The derivative of cos-1 (2x2 - 1) with respect to cos-1 x is

Solution 20

Correct option: (a)

Question 21

Solution 21

Correct option: (b)

Question 22

Solution 22

Correct option: (c)

Question 23

Solution 23

Correct option:(d)

Question 24

Solution 24

Correct option: (a)

Question 25

a. 1

b. 0

c. xl+m+n

d. none of these

Solution 25

Correct option: (b)

Question 26

a. 1

b.

c. 0

d. none of these

Solution 26

Correct option: (c)

Question 27

Solution 27

Correct option:(a)

Question 28

Solution 28

Correct option: (b)

Question 29

Solution 29

Correct option: (b)

Question 30

Solution 30

Correct option: (a)

Question 31

Solution 31

Correct option: (b)

Question 32

Solution 32

Correct option: (c)

## Chapter 11 - Differentiation Exercise Ex. 11.1

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4

Solution 4

Question 5
Solution 5

Question 6
Solution 6

Question 7

Solution 7

Question 8

Differentiate f(x)=x2ex from first principles.

Solution 8

Question 9

Solution 9

Question 10

Solution 10

## Chapter 11 - Differentiation Exercise Ex. 11.2

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15

Question 16

Solution 16

Question 17
Solution 17
Question 18
Solution 18

Question 19
Solution 19

Question 20
Solution 20
Question 21
Solution 21
Question 22
Solution 22
Question 23
Solution 23

Question 24

Solution 24

Question 25
Solution 25
Question 26
Solution 26

Question 27
Solution 27
Question 28
Solution 28
Question 29
Solution 29
Question 30
Solution 30
Question 31
Solution 31

Question 32
Solution 32
Question 33
Solution 33
Question 34
Solution 34
Question 35
Solution 35
Question 36
Solution 36
Question 37
Solution 37
Question 38
Solution 38
Question 39
Solution 39

Question 40
Solution 40
Question 41
Solution 41
Question 42
Solution 42
Question 43
Solution 43
Question 44
Solution 44
Question 45
Solution 45

Question 46
Solution 46
Question 47
Solution 47
Question 48
Solution 48

Question 49
Solution 49
Question 50
Solution 50
Question 51
Solution 51
Question 52
Solution 52
Question 53

Differentiate the following functions with respect to x:

Solution 53

Question 54
Solution 54
Question 55
Solution 55
Question 56
Solution 56
Question 57
Solution 57
Question 58
Solution 58
Question 59
Solution 59
Question 60
Solution 60
Question 61
Solution 61

Question 63
Solution 63

Question 64
Solution 64
Question 65
Solution 65

Question 66
Solution 66
Question 67
Solution 67
Question 68
Solution 68
Question 69
Solution 69
Question 70
Solution 70
Question 71
Solution 71
Question 72
Solution 72
Question 73
Solution 73

Question 74
Solution 74

Question 62

If  prove that

Solution 62

Given:

Differentiating w.r.t x, we get

Hence,

Question 75

If  find

Solution 75

Given:

Question 76

If  then find

Solution 76

Given:

## Chapter 11 - Differentiation Exercise Ex. 11.3

Question 1

Solution 1
Question 2
Solution 2

Question 3
Solution 3

Question 4
Solution 4
Question 5
Solution 5

Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9

Question 10
Solution 10
Question 11
Solution 11
Question 13
Solution 13

Question 14

Solution 14

Question 15
Solution 15

Question 16
Solution 16
Question 17

Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21
Solution 21
Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24
Question 25
Solution 25
Question 26
Solution 26

Question 27
Solution 27

Question 28
Solution 28
Question 29
Solution 29
Question 30
Solution 30
Question 31
Solution 31
Question 32
Solution 32
Question 33
Solution 33
Question 34
Solution 34
Question 35

Solution 35

Question 36
Solution 36
Question 37(i)
Solution 37(i)
Question 37(ii)
Solution 37(ii)

Question 38

show that dy/dx is independent of x.

Solution 38

Question 39
Solution 39

Question 40
Solution 40
Question 41
Solution 41

Question 42
Solution 42
Question 43
Solution 43

Question 44
Solution 44

Question 45

If y = tan-1

Solution 45

Question 46
Solution 46

Question 47

Solution 47

Question 12

Differentiate the following function with respect to x:

Solution 12

Let

Question 48

If  then find

Solution 48

Given: ………. (i)

Let

From (i), we get

## Chapter 11 - Differentiation Exercise Ex. 11.4

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9

Question 10
Solution 10
Question 11
Solution 11

Question 12
Solution 12
Question 13

Solution 13

Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19

Question 20
Solution 20
Question 21
Solution 21
Question 22
Solution 22

Question 23
Solution 23
Question 24
Solution 24
Question 25

Solution 25

Question 26

Solution 26

Question 27

.

Solution 27

Question 28

Solution 28

Question 30

Solution 30

Question 31

Solution 31

Question 29

If find  at x =1,

Solution 29

Given:

Differentiating w.r.t x. we get

When x =1 and  we get

## Chapter 11 - Differentiation Exercise Ex. 11.5

Question 1

Solution 1

Question 2

Solution 2

Question 3
Solution 3

Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17

Question 18(i)
Solution 18(i)

Question 18(ii)
Solution 18(ii)
Question 18(iii)
Solution 18(iii)
Question 18(iv)
Solution 18(iv)
Question 18(v)
Solution 18(v)
Question 18(vi)
Solution 18(vi)
Question 18(vii)
Solution 18(vii)
Question 18(viii)
Solution 18(viii)
Question 19
Solution 19
Question 20
Solution 20
Question 21
Solution 21

Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24

Question 25

Solution 25

Question 26
Solution 26
Question 27
Solution 27
Question 29(i)
Solution 29(i)
Question 29(ii)
Solution 29(ii)
Question 30
Solution 30
Question 31
Solution 31
Question 32
Solution 32
Question 33
Solution 33
Question 34
Solution 34

Question 35
Solution 35
Question 36

Solution 36

Question 37

Solution 37

Question 38
Solution 38

Question 39
Solution 39

Question 40
Solution 40
Question 41
Solution 41
Question 42
Solution 42

Question 43
Solution 43
Question 44
Solution 44
Question 45
Solution 45
Question 46
Solution 46
Question 47
Solution 47
Question 48
Solution 48
Question 49

Solution 49

Question 50
Solution 50
Question 51
Solution 51
Question 52
Solution 52
Question 53
Solution 53

Question 54
Solution 54
Question 55
Solution 55

Question 56
Solution 56
Question 57

Solution 57

Question 58

Solution 58

Question 59

Solution 59

Question 60

Solution 60

Question 61

Solution 61

Question 28

Find  when

Solution 28

Given:

Let

Differentiating 'u' w.r.t x, we get

Differentiating 'v' w.r.t x, we get

From (i), (ii) and (iii), we get

Question 62

If  find

Solution 62

Given:

Let

Taking log on both the sides of equation (i), we get

Taking log on both the sides of equation (ii), we get

Differentiating (iii) w.r.t x, we get

Using (iv) and (v), we have

Question 1
Solution 1

Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5

Question 6
Solution 6
Question 7
Solution 7

Question 8

Solution 8

## Chapter 11 - Differentiation Exercise Ex. 11.7

Question 1
Solution 1
Question 2
Solution 2
Question 3

Solution 3

Question 4
Solution 4
Question 5
Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8
Solution 8

Question 9

Solution 9
Question 10
Solution 10

Question 11
Solution 11

Question 12
Solution 12

Question 13
Solution 13

Question 14
Solution 14

Question 15
Solution 15
Question 16
Solution 16
Question 17

Solution 17

Question 18

Solution 18

Question 19
Solution 19
Question 20

Solution 20

Question 21
Solution 21

Question 22

Solution 22

Question 23
Solution 23
Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

If  find  when

Solution 29

Given:

Differentiate 'x' w.r.t , we get

Differentiate 'y' w.r.t , we get

Dividing (ii) by (i), we get

At

## Chapter 11 - Differentiation Exercise Ex. 11.8

Question 2
Solution 2
Question 3
Solution 3

Question 4(i)
Solution 4(i)
Question 4(ii)

Solution 4(ii)

Question 5(i)

Solution 5(i)

Question 5(ii)
Solution 5(ii)
Question 5(iii)
Solution 5(iii)

Question 6
Solution 6

Question 7(i)
Solution 7(i)

Question 7(ii)
Solution 7(ii)

Question 8
Solution 8
Question 9
Solution 9

Question 10
Solution 10

Question 11
Solution 11

Question 12

Solution 12

Question 13
Solution 13

Question 14
Solution 14

Question 15
Solution 15

Question 16
Solution 16

Question 17
Solution 17

Question 18
Solution 18

Question 19
Solution 19

Question 20
Solution 20

Question 1

Differentiate  with respect to

Solution 1

We need to find

Let

So, we need to find

Question 21

Differentiate  with respect to

Solution 21

We need to find

Let

Differentiating 'u' and 'v' w.r.t x, we get

Dividing (i) by (ii), we get

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