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

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

begin mathsize 12px style If space straight y equals sin to the power of negative 1 end exponent open parentheses sin space straight x close parentheses comma negative straight pi over 2 less or equal than straight x less or equal than straight pi over 2. Then comma space write space the space value space of space dy over dx for space straight x element of open parentheses negative straight pi over 2 comma straight pi over 2 close parentheses. end style

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

begin mathsize 12px style If space straight f left parenthesis straight x right parenthesis space equals space log open curly brackets fraction numerator straight u open parentheses straight x close parentheses over denominator straight v open parentheses straight x close parentheses end fraction close curly brackets comma space straight u open parentheses 1 close parentheses equals straight v open parentheses 1 close parentheses space and space straight u apostrophe open parentheses 1 close parentheses equals straight v apostrophe open parentheses 1 close parentheses equals 2 comma space then space find space the space value space of space straight f apostrophe open parentheses 1 close parentheses. end style

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

begin mathsize 12px style If space straight f left parenthesis straight x right parenthesis equals left parenthesis straight x to the power of straight l over straight x to the power of straight m right parenthesis to the power of straight l plus straight m end exponent left parenthesis straight x to the power of straight m over straight x to the power of straight n right parenthesis to the power of straight m plus straight n end exponent open parentheses straight x to the power of straight n over straight x to the power of straight l close parentheses to the power of straight n plus straight l end exponent comma space then space straight f apostrophe space left parenthesis straight x right parenthesis space is space equal space to end style

a. 1

b. 0

c. xl+m+n

d. none of these


Solution 25

Correct option: (b)

  

Question 26

begin mathsize 12px style If comma space straight y equals fraction numerator 1 over denominator 1 plus straight x to the power of straight a minus straight b end exponent plus straight x to the power of straight c minus straight b end exponent end fraction plus fraction numerator 1 over denominator 1 plus straight x to the power of straight b minus straight c end exponent plus straight x to the power of straight a minus straight c end exponent end fraction plus fraction numerator 1 over denominator 1 plus straight x to the power of straight b minus straight a end exponent plus straight x to the power of straight c minus straight a end exponent end fraction comma space then
dy over dx space is space equal space to end style


a. 1

b. begin mathsize 12px style left parenthesis straight a space plus space straight b space plus space straight c right parenthesis open curly brackets blank to the power of straight x to the power of open parentheses straight a space plus space straight b space plus space straight c space minus space 1 close parentheses end exponent end exponent close curly brackets end style

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


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

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

begin mathsize 12px style Differentiate space straight y equals sin to the power of negative 1 end exponent open parentheses fraction numerator straight x plus square root of 1 minus straight x squared end root over denominator square root of 2 end fraction close parentheses comma fraction numerator negative 1 over denominator square root of 2 end fraction less than straight x less than fraction numerator 1 over denominator square root of 2 end fraction end style

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

begin mathsize 12px style If space straight y equals sin to the power of negative 1 end exponent open parentheses fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction close parentheses plus sec to the power of negative 1 end exponent open parentheses fraction numerator 1 plus straight x squared over denominator 1 minus straight x squared end fraction close parentheses comma space 0 less than straight x less than 1 comma space prove space that space dy over dx equals fraction numerator 4 over denominator 1 plus straight x squared end fraction. end style

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 begin mathsize 12px style If space straight y space equals space tan to the power of negative 1 end exponent open parentheses fraction numerator square root of 1 plus straight x end root minus space square root of 1 minus straight x end root over denominator square root of 1 plus straight x end root plus space square root of 1 minus straight x end root end fraction close parentheses comma space find space dy over dx end style

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

begin mathsize 12px style If space straight y square root of 1 minus straight x squared end root plus space straight x square root of 1 minus straight y squared end root equals 1 comma space prove space that space dy over dx equals negative square root of fraction numerator 1 minus straight y squared over denominator 1 minus straight x squared end fraction end root end style

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

begin mathsize 12px style dy over dx plus space straight e to the power of open curly brackets straight y minus straight x close curly brackets end exponent equals 0 end style.

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

begin mathsize 12px style If space straight x to the power of straight x plus straight y to the power of straight x equals 1 comma space prove space that space dy over dx equals negative open curly brackets fraction numerator straight x to the power of straight x open parentheses 1 plus logx close parentheses plus straight y to the power of straight x cross times space logy over denominator straight x space cross times space straight y to the power of open parentheses straight x minus 1 close parentheses end exponent end fraction close curly brackets end style

Solution 36

Question 37

begin mathsize 12px style If space straight x to the power of straight y space cross times space straight y to the power of straight x equals 1 comma space prove space that space dy over dx equals negative fraction numerator straight y open parentheses straight y plus xlogy close parentheses over denominator straight x open parentheses ylogx plus straight x close parentheses end fraction end style

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

begin mathsize 12px style If space straight y equals 1 plus fraction numerator straight alpha over denominator open parentheses begin display style 1 over straight x end style minus straight alpha close parentheses end fraction plus fraction numerator straight beta divided by straight x over denominator open parentheses begin display style 1 over straight x end style minus straight alpha close parentheses open parentheses begin display style 1 over straight x end style minus straight beta close parentheses end fraction plus fraction numerator straight gamma divided by straight x squared over denominator open parentheses begin display style 1 over straight x end style minus straight alpha close parentheses open parentheses begin display style 1 over straight x end style minus straight beta close parentheses open parentheses begin display style 1 over straight x end style minus straight gamma close parentheses end fraction comma space find space dy over dx. end style

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

  

Chapter 11 - Differentiation Exercise Ex. 11.6

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

i f space y space equals space open parentheses cos x close parentheses to the power of open parentheses cos x close parentheses to the power of open parentheses cos x close parentheses to the power of negative y end exponent end exponent end exponent comma space p r o v e space t h a t space fraction numerator d y over denominator d x end fraction equals negative fraction numerator y squared tan x over denominator open parentheses 1 minus y log cos x close parentheses end fraction.

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

begin mathsize 12px style If space straight x equals straight a open parentheses straight t plus 1 over straight t close parentheses space and space straight y equals straight a open parentheses straight t minus 1 over straight t close parentheses comma space prove space that space dy over dx equals straight x over straight y. end style

Solution 17

Question 18

begin mathsize 12px style If space straight x equals sin to the power of negative 1 end exponent open parentheses fraction numerator 2 straight t over denominator 1 plus straight t squared end fraction close parentheses space and space straight y space equals space tan to the power of negative 1 end exponent open parentheses fraction numerator 2 straight t over denominator 1 plus straight t squared end fraction close parentheses. space minus 1 less than straight t less than 1 comma space prove space that space dy over dx equals 1. end style

Solution 18

Question 19
Solution 19
Question 20

Solution 20

Question 21
Solution 21

Question 22

begin mathsize 12px style Find space dy over dx comma space if space straight y equals 12 open parentheses 1 minus cost close parentheses comma straight x equals 10 open parentheses straight t minus sint close parentheses. end style

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)

begin mathsize 12px style Differentiate space sin to the power of negative 1 end exponent square root of 1 minus straight x squared end root with space respect space to space cos to the power of negative 1 end exponent straight x comma space if
straight x space element of space open parentheses negative 1 comma space 0 close parentheses end style

Solution 4(ii)

Question 5(i)

begin mathsize 12px style Differentiate space sin to the power of negative 1 end exponent open parentheses 4 straight x square root of 1 minus 4 straight x squared end root close parentheses space space with space space respect space to space square root of 1 minus 4 straight x squared end root comma space if
straight x space element of open parentheses fraction numerator 1 over denominator negative 2 square root of 2 end fraction comma fraction numerator 1 over denominator 2 square root of 2 end fraction close parentheses end style

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