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Continuity And Differentiability

Question
CBSEENMA12035049
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Differentiate xtan x+(sin x)cos x w.r.t.x.

Solution
Let space straight y equals straight x to the power of tan space straight x end exponent space plus left parenthesis sin space straight x right parenthesis to the power of cos space straight x end exponent Put space straight x to the power of tan space straight x end exponent equals straight u comma space left parenthesis sin space straight x right parenthesis to the power of cos space straight x end exponent equals straight v therefore space straight y equals straight u plus straight v therefore space dy over dx equals du over dx plus dv over dx space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis space space space space space straight u equals space straight x to the power of tan space straight x end exponent therefore space log space straight u equals log space straight x to the power of tan space straight x end exponent equals tan space straight x. log space straight x therefore space 1 over straight u du over dx equals fraction numerator tan space straight x over denominator straight x end fraction plus sec squared straight x. log space straight x therefore space du over dx equals space straight x to the power of tan space straight x end exponent open square brackets fraction numerator tan space straight x over denominator straight x end fraction plus sec squared straight x. log space straight x close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis
space space space space space space space space space space space space space space space space straight v equals left parenthesis sin space straight x right parenthesis to the power of cos space straight x end exponent therefore space space space space space log space straight v equals log left parenthesis sin space straight x right parenthesis to the power of cos space straight x end exponent equals cos space straight x. log space sin space straight x therefore space 1 over straight v dv over dx equals cos space straight x. fraction numerator 1 over denominator sin space straight x end fraction. cos space straight x minus sin space straight x. log space sin space straight x therefore space dv over dx equals left parenthesis sin space straight x right parenthesis to the power of cos space straight x end exponent left square bracket cos space straight x space cot space straight x minus sin space straight x. log space sin space straight x right square bracket space space space space space space space space space space space space space space space space space space space... left parenthesis 3 right parenthesis From space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis space and space left parenthesis 3 right parenthesis comma space we space get. dy over dx equals straight x to the power of tan space straight x end exponent open square brackets fraction numerator tan space straight x over denominator straight x end fraction plus sec squared straight x. log space straight x close square brackets left parenthesis sin space straight x right parenthesis to the power of cos space straight x end exponent left square bracket cos space xcot space straight x minus sin space straight x. log space straight x. sin space straight x right square bracket

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