Find the derevative of f(x)=tanx^1/3 using fundamental method.

Asked by Balbir | 10th Feb, 2017, 10:06: AM

Expert Answer:

begin mathsize 16px style The space fundamental space Method space is space also space called space differentiation space from space First space Principles space which space is space given space by
straight f apostrophe left parenthesis straight x right parenthesis equals limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis straight x plus straight h right parenthesis minus straight f left parenthesis straight x right parenthesis over denominator straight h end fraction
So comma space we space will space use space the space above space definition space to space differentiate.
Given space straight f left parenthesis straight x right parenthesis equals cube root of tan space straight x end root space space and space space space straight f left parenthesis straight x right parenthesis space equals space cube root of tan space left parenthesis straight x plus straight h right parenthesis end root
straight d over dx left parenthesis straight f left parenthesis straight x right parenthesis right parenthesis equals limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis straight x plus straight h right parenthesis minus straight f left parenthesis straight x right parenthesis over denominator straight h end fraction
space space space space space space space space space space space space space space equals limit as straight h rightwards arrow 0 of fraction numerator cube root of tan space left parenthesis straight x plus straight h right parenthesis end root minus cube root of tan space straight x end root space over denominator straight h end fraction
space space space space space space space space space space space space space space equals limit as straight h rightwards arrow 0 of fraction numerator open curly brackets cube root of tan space left parenthesis straight x plus straight h right parenthesis end root close curly brackets cubed minus open curly brackets cube root of tan space straight x end root close curly brackets cubed over denominator space space space straight h space space space open square brackets tan space to the power of 2 divided by 3 end exponent left parenthesis straight x plus straight h right parenthesis plus tan space to the power of 1 divided by 3 end exponent left parenthesis straight x plus straight h right parenthesis tan space to the power of 1 divided by 3 end exponent left parenthesis straight x right parenthesis plus tan space to the power of 2 divided by 3 end exponent left parenthesis straight x right parenthesis close square brackets end fraction space space...... open square brackets Since space straight a minus straight b equals fraction numerator straight a cubed minus straight b cubed over denominator straight a squared plus ab plus straight b squared end fraction close square brackets
space space space space space space space space space space space space space space equals limit as straight h rightwards arrow 0 of fraction numerator tan space left parenthesis straight x plus straight h right parenthesis minus tan space straight x over denominator straight h end fraction fraction numerator 1 over denominator open square brackets tan space to the power of 2 divided by 3 end exponent left parenthesis straight x plus straight h right parenthesis plus tan space to the power of 1 divided by 3 end exponent left parenthesis straight x plus straight h right parenthesis tan space to the power of 1 divided by 3 end exponent left parenthesis straight x right parenthesis plus tan space to the power of 2 divided by 3 end exponent left parenthesis straight x right parenthesis close square brackets end fraction
space space space space space space space space space space space space space space equals limit as straight h rightwards arrow 0 of fraction numerator sin space straight h over denominator straight h space cos left parenthesis straight x plus straight h right parenthesis space cosx end fraction fraction numerator 1 over denominator open square brackets tan space to the power of 2 divided by 3 end exponent left parenthesis straight x plus straight h right parenthesis plus tan space to the power of 1 divided by 3 end exponent left parenthesis straight x plus straight h right parenthesis tan space to the power of 1 divided by 3 end exponent left parenthesis straight x right parenthesis plus tan space to the power of 2 divided by 3 end exponent left parenthesis straight x right parenthesis close square brackets end fraction...... open square brackets Since space tanA minus tanB equals fraction numerator sin space left parenthesis straight A minus straight B right parenthesis over denominator cosA space cosB end fraction close square brackets
space space space space space space space space space space space space space space equals limit as straight h rightwards arrow 0 of fraction numerator sin space straight h over denominator straight h space end fraction fraction numerator 1 over denominator space cos left parenthesis straight x plus straight h right parenthesis space cosx end fraction fraction numerator 1 over denominator open square brackets tan space to the power of 2 divided by 3 end exponent left parenthesis straight x plus straight h right parenthesis plus tan space to the power of 1 divided by 3 end exponent left parenthesis straight x plus straight h right parenthesis tan space to the power of 1 divided by 3 end exponent left parenthesis straight x right parenthesis plus tan space to the power of 2 divided by 3 end exponent left parenthesis straight x right parenthesis close square brackets end fraction
space space space space space space space space space space space space space space equals left parenthesis 1 right parenthesis fraction numerator 1 over denominator space cos squared straight x space end fraction fraction numerator 1 over denominator open square brackets tan space to the power of 2 divided by 3 end exponent left parenthesis straight x right parenthesis plus tan space to the power of 1 divided by 3 end exponent left parenthesis straight x right parenthesis tan space to the power of 1 divided by 3 end exponent left parenthesis straight x right parenthesis plus tan space to the power of 2 divided by 3 end exponent left parenthesis straight x right parenthesis close square brackets end fraction
space space space space space space space space space space space space space space equals left parenthesis 1 right parenthesis fraction numerator 1 over denominator space cos squared straight x space end fraction fraction numerator 1 over denominator open square brackets tan space to the power of 2 divided by 3 end exponent left parenthesis straight x right parenthesis plus tan space to the power of 2 divided by 3 end exponent left parenthesis straight x right parenthesis plus tan space to the power of 2 divided by 3 end exponent left parenthesis straight x right parenthesis close square brackets end fraction
space space space space space space space space space space space space space space equals fraction numerator sec squared straight x over denominator 3 tan space to the power of 2 divided by 3 end exponent left parenthesis straight x right parenthesis end fraction

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Answered by Rebecca Fernandes | 10th Feb, 2017, 11:17: AM