Chapter 11 Differentiation 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 styleSolution 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

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