Chapter 30 : Derivatives - Rd Sharma Solutions for Class 11-science Maths CBSE

Chapter 30 - Derivatives Exercise Ex. 30.1

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

Solution 8

Question 9

Solution 9

Question 10
Solution 10

Chapter 30 - Derivatives Exercise Ex. 30.2

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

f left parenthesis x right parenthesis equals a to the power of square root of x end exponent equals e to the power of square root of x log a end exponent

f to the power of comma left parenthesis x right parenthesis equals limit as h rightwards arrow 0 of fraction numerator f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis over denominator h end fraction
space space space space space space space space equals limit as h rightwards arrow 0 of fraction numerator e to the power of square root of x plus h end root log a end exponent minus e to the power of square root of x log a end exponent over denominator h end fraction
space space space space space space space space equals limit as h rightwards arrow 0 of e to the power of square root of x log a end exponent fraction numerator e to the power of square root of x plus h end root log a minus square root of x log a end exponent minus 1 over denominator h end fraction
space space space space space space space space equals limit as h rightwards arrow 0 of e to the power of square root of x log a end exponent fraction numerator e to the power of open parentheses square root of x plus h end root minus square root of x close parentheses log a end exponent minus 1 over denominator h end fraction
text Multiply   numerator   and   denominator   by end text space open parentheses square root of x plus h end root minus square root of x close parentheses log a
f to the power of comma left parenthesis x right parenthesis space equals limit as h rightwards arrow 0 of e to the power of square root of x log a end exponent fraction numerator e to the power of open parentheses square root of x plus h end root minus square root of x close parentheses log a end exponent minus 1 over denominator h open parentheses square root of x plus h end root minus square root of x close parentheses log a end fraction open parentheses square root of x plus h end root minus square root of x close parentheses log a
space space space space space space space space equals e to the power of square root of x log a end exponent limit as h rightwards arrow 0 of fraction numerator e to the power of open parentheses square root of x plus h end root minus square root of x close parentheses log a end exponent minus 1 over denominator open parentheses square root of x plus h end root minus square root of x close parentheses log a end fraction limit as h rightwards arrow 0 of log a fraction numerator open parentheses square root of x plus h end root minus square root of x close parentheses over denominator h end fraction
space space space space space space space space space equals e to the power of square root of x log a end exponent limit as h rightwards arrow 0 of log a fraction numerator open parentheses square root of x plus h end root minus square root of x close parentheses over denominator h end fraction

text Multiply   numerator   and   denominator   by end text space open parentheses square root of x plus h end root plus square root of x close parentheses
space f to the power of comma left parenthesis x right parenthesis equals e to the power of square root of x log a end exponent limit as h rightwards arrow 0 of log a fraction numerator open parentheses square root of x plus h end root minus square root of x close parentheses over denominator h open parentheses square root of x plus h end root plus square root of x close parentheses end fraction open parentheses square root of x plus h end root plus square root of x close parentheses
equals e to the power of square root of x log a end exponent limit as h rightwards arrow 0 of log a fraction numerator h over denominator h open parentheses square root of x plus h end root plus square root of x close parentheses end fraction
equals e to the power of square root of x log a end exponent fraction numerator log a over denominator 2 square root of x end fraction
equals fraction numerator a square root of blank to the power of x end root over denominator 2 square root of x end fraction log subscript e a

 

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

Differentiate -x using first principles.

Solution 47

  

Question 48

Differentiate (-x)-1 using first principles.

Solution 48

  

Question 49

Differentiate sin(x + 1) using first principles.

Solution 49

  

Question 50

Differentiate cos  using first principles.

Solution 50

  

Chapter 30 - Derivatives Exercise Ex. 30.3

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 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Defferentiate f (x) = log open parentheses fraction numerator 1 over denominator square root of straight x end fraction close parentheses plus 5 straight x to the power of straight a minus 3 straight a to the power of straight x plus 3 square root of straight x squared end root plus 6 fourth root of straight x to the power of negative 3 end exponent end root with space respect space to space straight x.

Solution 16

fraction numerator d over denominator a x end fraction open curly brackets log open parentheses fraction numerator 1 over denominator square root of x end fraction close parentheses plus 5 x to the power of a minus 3 a to the power of x plus 3 root of x squared end root plus 6 4 root of x to the power of minus 3 end exponent end root close curly brackets
equals fraction numerator d over denominator a x end fraction log open parentheses fraction numerator 1 over denominator square root of x end fraction close parentheses plus 5 fraction numerator d over denominator a x end fraction open parentheses x to the power of a close parentheses minus 3 open parentheses a to the power of x close parentheses plus fraction numerator d over denominator a x end fraction open parentheses 3 root of x squared end root close parentheses plus 6 fraction numerator d over denominator a x end fraction open parentheses 4 root of x to the power of minus 3 end exponent end root close parentheses
equals fraction numerator minus 1 over denominator 2 end fraction 1 over x plus 5 a x to the power of a minus 1 end exponent minus 3 a to the power of x log a plus fraction numerator 2 x to the power of begin display style bevelled fraction numerator minus 1 over denominator 3 end fraction end style end exponent over denominator 3 end fraction plus 6 x to the power of bevelled fraction numerator minus 7 over denominator 4 end fraction end exponent open parentheses bevelled fraction numerator minus 3 over denominator 4 end fraction close parentheses
equals fraction numerator minus 1 over denominator 2 x end fraction plus 5 a x to the power of a minus 1 end exponent minus 3 a to the power of x log a plus fraction numerator 2 x to the power of begin display style bevelled fraction numerator minus 1 over denominator 3 end fraction end style end exponent over denominator 3 end fraction minus 9 over 2 x to the power of bevelled fraction numerator minus 7 over denominator 4 end fraction end exponent

 

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

Chapter 30 - Derivatives Exercise Ex. 30.4

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

Differentiate the following functions with respect to x:

  

Solution 25

  

Question 26

Differentiate the following functions with respect to x:

(ax + b)n (cx + d)m

Solution 26

  

Question 27

Differentiate in two ways, using product rule and otherwise, the function (1 + 2 tan x) (5 + 4 cos x). Verify that the answer are the same.

Solution 27

  

Question 28

Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.

(3x2 + 2)2

Solution 28

  

Question 29

Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.

(x + 2)(x + 3)

Solution 29

  

Question 30

Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.

(3 sec x - 4 cosec x) (-2 sin x + 5 cos x)

Solution 30

  

Chapter 30 - Derivatives Exercise Ex. 30.5

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

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

text Differentiate end text fraction numerator x to the power of n over denominator sin x end fraction text   with   respect   to   x. end text

Solution 28

fraction numerator d over denominator d x end fraction open parentheses fraction numerator x to the power of n over denominator sin x end fraction close parentheses
equals x to the power of n fraction numerator d over denominator d x end fraction open parentheses sin x close parentheses to the power of minus 1 end exponent plus fraction numerator 1 over denominator sin x end fraction fraction numerator d over denominator d x end fraction open parentheses x to the power of n close parentheses
equals x to the power of n fraction numerator minus 1 over denominator sin squared x end fraction plus fraction numerator 1 over denominator sin x end fraction n x to the power of n minus 1 end exponent
equals fraction numerator sin x left parenthesis n x to the power of n minus 1 end exponent right parenthesis minus x to the power of n left parenthesis cos x right parenthesis over denominator sin squared x end fraction

Question 29

Differentiate the following functions with respect to x:

 fraction numerator a x plus b over denominator p x squared plus q x plus r end fraction

Solution 29

Question 30

Differentiate the following functions with respect to x:

fraction numerator 1 over denominator a x squared plus b x plus c end fraction

Solution 30

Syntax error from line 1 column 1707 to line 1 column 1712. Unexpected '</mi>'.

Chapter 30 - Derivatives Exercise Ex. 30VSAQ

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

text for   x = 2 end text

open vertical bar x minus 1 close vertical bar equals left parenthesis x minus 1 right parenthesis space space space text and     end text open vertical bar x minus 3 close vertical bar equals minus open parentheses x minus 3 close parentheses

fraction numerator d over denominator d x end fraction open parentheses open vertical bar x minus 1 close vertical bar plus open vertical bar x minus 3 close vertical bar close parentheses
equals fraction numerator d over denominator d x end fraction open parentheses open vertical bar x minus 1 close vertical bar close parentheses plus fraction numerator d over denominator d x end fraction open parentheses open vertical bar x minus 3 close vertical bar close parentheses
equals fraction numerator d over denominator d x end fraction open parentheses x minus 1 close parentheses plus fraction numerator d over denominator d x end fraction open parentheses minus left parenthesis x minus 3 right parenthesis close parentheses
equals fraction numerator d over denominator d x end fraction open parentheses x minus 1 close parentheses plus fraction numerator d over denominator d x end fraction open parentheses 3 minus x right parenthesis close parentheses
equals 1 minus 1
equals 0

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

text We   have end text
y equals 1 plus x plus x squared plus x cubed plus horizontal ellipsis
space space space equals open parentheses 1 minus x close parentheses to the power of minus 1 end exponent

fraction numerator d y over denominator d x end fraction equals fraction numerator d over denominator d x end fraction open parentheses 1 minus x close parentheses to the power of minus 1 end exponent
space space space space space space space space equals open parentheses minus 1 close parentheses open parentheses 1 minus x close parentheses to the power of minus 2 end exponent cross times open parentheses minus 1 close parentheses
space space space space space space space space equals open parentheses 1 minus x close parentheses to the power of minus 2 end exponent
space space space space space space space space equals 1 over open parentheses 1 minus x close parentheses squared

Question 14

Solution 14

Loading...