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

Differentiation Exercise Ex. 11.1

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5


Solution 6


Solution 7

Solution 8

Solution 9

Solution 10


Differentiation Exercise Ex. 11.2

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15


Solution 16


T h u s comma space fraction numerator d y over denominator d x end fraction equals fraction numerator 1 over denominator cos squared x end fraction plus fraction numerator sin x over denominator cos squared x end fraction rightwards double arrow fraction numerator d y over denominator d x end fraction equals s e c squared x plus tan x s e c x rightwards double arrow fraction numerator d y over denominator d x end fraction equals s e c x open square brackets tan x plus s e c x close square brackets

Solution 17

Solution 18


Solution 19


Solution 20

Solution 21

Solution 22

Solution 23


Solution 24

Solution 25

Solution 26


Solution 27

Solution 28

Solution 29

Solution 30

Solution 31


Solution 32

Solution 33

Solution 34

Solution 35

Solution 36

Solution 37

Solution 38

Solution 39


Solution 40

Solution 41

Solution 42

Solution 43

Solution 44

Solution 45



Solution 46

Solution 47

Solution 48


Solution 49

Solution 50

Solution 51

Solution 52

Solution 53

Solution 54

Solution 55

Solution 56

Solution 57

Solution 58

Solution 59

Solution 60

Solution 61


Solution 62

Given:

Differentiating w.r.t x, we get

 

Hence,  

Solution 63


Solution 64

Solution 65


Solution 66

Solution 67

Solution 68

Solution 69

Solution 70

Solution 71

Solution 72

Solution 73


Solution 74


Solution 75

Given:

 

 

Solution 76

Given:

 

Differentiation Exercise Ex. 11.3

Solution 1

Solution 2


Solution 3


Solution 4

Solution 5


Solution 6

Solution 7

Solution 8

Solution 9


Solution 10

Solution 11

Solution 12

Let

 

Solution 13


Solution 14

Solution 15


Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26


Solution 27


Solution 28

Solution 29

Solution 30

Solution 31

Solution 32

Solution 33

Solution 34

Solution 35

Solution 36

Solution 37(i)

Solution 37(ii)


Solution 38

Solution 39


Solution 40

Solution 41


Solution 42

Solution 43


Solution 44


Solution 45

Solution 46


Solution 47

Solution 48

Given: ………. (i)

Let

From (i), we get

 

Differentiation Exercise Ex. 11.4

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9


Solution 10

Solution 11


Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19


Solution 20

Solution 21

Solution 22


Solution 23

Solution 24

Solution 25

Solution 26

Solution 27

Solution 28

Solution 29

Given:

Differentiating w.r.t x. we get

 

When x =1 and  we get

 

Solution 30


Solution 31

Differentiation Exercise Ex. 11.5

Solution 1


Solution 2


Solution 3


Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17


Solution 18(i)


Solution 18(ii)

Solution 18(iii)

Solution 18(iv)

Solution 18(v)

Solution 18(vi)

Solution 18(vii)

Solution 18(viii)

Solution 19

Solution 20

Solution 21


Solution 22

Solution 23

Solution 24


Solution 25


Solution 26

Solution 27

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

 

Solution 29(i)

Solution 29(ii)

Solution 30

Solution 31

Solution 32

Solution 33

Solution 34


Solution 35

Solution 36

Solution 37

Solution 38


Solution 39


Solution 40

Solution 41

Solution 42


Solution 43

Solution 44

Solution 45

Solution 46

Solution 47

Solution 48

Solution 49


Solution 50

Solution 51

Solution 52

Solution 53


Solution 54

Solution 55


Solution 56

Solution 57

Solution 58

Solution 59

Solution 60

Solution 61

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

 

Differentiation Exercise Ex. 11.6

Solution 1


Solution 2

Solution 3

Solution 4

Solution 5


Solution 6

Solution 7


Solution 8

Differentiation Exercise Ex. 11.7

Solution 1

Solution 2

Solution 3


Solution 4

Solution 5


Solution 6


Solution 7


Solution 8


Solution 9

Solution 10


Solution 11


Solution 12



Solution 13


Solution 14



Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20


Solution 21


Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

Solution 27

Solution 28

Solution 29

Given:

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

 

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

 

Dividing (ii) by (i), we get

 

At

 

Differentiation Exercise Ex. 11.8

Solution 1

We need to find

Let

So, we need to find

 

Solution 2

Solution 3


Solution 4(i)

Solution 4(ii)

Solution 5(i)

Solution 5(ii)

Solution 5(iii)


Solution 6



Solution 7(i)



Solution 7(ii)



Solution 8

Solution 9


Solution 10


Solution 11


Solution 12


Solution 13



Solution 14


Solution 15


Solution 16


Solution 17


Solution 18


Solution 19


Solution 20



Solution 21

We need to find

Let

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

 

Dividing (i) by (ii), we get

 

Differentiation Exercise MCQ

Solution 1

Correct option: (d)

 

 

Solution 2

Correct option: (c)

 

Solution 3

Correct option: (a)

 

Solution 4

Correct option: (d)

 

Solution 5

Correct option: (d)

 

Solution 6

Correct option: (a)

 

Solution 7

Correct option: (d)

 

Solution 8

Correct option: (c)

 

 

Solution 9

Correct option: (d)

 

Solution 10

Correct option:(a)

 

Solution 11

Correct option: (a)

 

Solution 12

Correct option: (c)

 

Solution 13

Correct option: (d)

 

Solution 14

Correct option: (d)

 

Solution 15

Correct option: (b)

 

Solution 16

Correct option: (a)

 

Solution 17

Correct option: (d)

 

Solution 18

Correct option: (a)

 

Solution 19

Correct option: (b)

 

Solution 20

Correct option: (a)

 

Solution 21

Correct option: (b)

 

Solution 22

Correct option: (c)

 

Solution 23

Correct option:(d)

 

Solution 24

Correct option: (a)

 

Solution 25

Correct option: (b)

 

Solution 26

Correct option: (c)

 

Solution 27

Correct option:(a)

 

Solution 28

Correct option: (b)

 

Solution 29

Correct option: (b)

 

Solution 30

Correct option: (a) 

 

 

Solution 31

Correct option: (b)

 

Solution 32

Correct option: (c)

 

Solution 33

Given:

 

Differentiating w.r.t x, we get

 

Differentiation Exercise Ex. 11VSAQ

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

 

Solution 25

Solution 26

Solution 27

Solution 28

Solution 29

Given:

 

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

 

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