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Class 11-commerce RD SHARMA Solutions Maths Chapter 30: Derivatives

Derivatives Exercise Ex. 30.1

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7 (i)

Solution 7 (ii)

Solution 7 (iii)

Solution 7(iv)

Derivatives Exercise Ex. 30.2

Solution 1(i)

Solution 1 (ii)

Solution 1 (iii)

Solution 1 (iv)

Solution 1 (v)

Solution 1 (vi)

Solution 1 (vii)

Solution 1 (viii)

Solution 1 (ix)

Solution 1 (x)

Solution 1 (xi)

Solution 1 (xii)

Solution 1 (xiii)

Solution 1 (xiv)

Solution 2 (i)

Solution 2 (ii)

Solution 2 (iii)

Solution 2 (iv)

Solution 3 (vii)

Solution 3 (viii)

Solution 3 (ix)

Solution 3 (x)

Solution 3 (xi)

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

 

Solution 3 (xii)

Solution 2(ix)

Solution 2(x)

Solution 2(xi)

Solution 3 (i)

Solution 3 (ii)

Solution 3 (iii)

Solution 3 (iv)

Solution 3 (v)

Solution 3 (vi)

Solution 4 (i)

Solution 4 (ii)

Solution 4 (iii)

Solution 4 (iv)

Solution 5(i)

Solution 5 (ii)

Solution 5 (iii)

Solution 5(iv)

Solution 6 (i)

Solution 6 (ii)

Solution 6(iii)

Solution 6(iv)

Solution 2(v)

 

Solution 2(vi)

 

Solution 2(vii)

 

Solution 2(viii)

 

Solution 1(xv)

Derivatives Exercise Ex. 30.3

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

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

 

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

Derivatives Exercise Ex. 30.4

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(i)

 

Solution 28(ii)

 

Solution 28(iii)

 

Derivatives Exercise Ex. 30.5

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

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

Solution 27

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

Solution 29

Solution 30

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Derivatives Exercise Ex. 30VSAQ

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

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

Solution 9

Solution 10

Solution 11

Solution 12

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

Solution 14

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