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Continuity And Differentiability

Question
CBSEENMA12035053
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If space straight y equals left parenthesis straight x right parenthesis to the power of cos space straight x end exponent plus left parenthesis cos space straight x right parenthesis to the power of sin space straight x end exponent comma space find space dy over dx.

Solution
Here space straight y equals left parenthesis straight x right parenthesis to the power of cos space straight x end exponent plus left parenthesis cos space straight x right parenthesis to the power of sin space straight x end exponent Put space left parenthesis straight x right parenthesis to the power of cos space straight x end exponent equals straight u comma space left parenthesis cos space straight x right parenthesis to the power of sin 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 space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis Now space space space straight u equals straight x to the power of cos space straight x end exponent therefore space log space straight u equals log space straight x to the power of cos space straight x end exponent therefore space log space straight u equals cos space straight x. log space straight x therefore space 1 over straight u du over dx equals cos space straight x.1 over straight x plus log space straight x. left parenthesis negative sin space straight x right parenthesis therefore space du over dx equals straight u open square brackets fraction numerator cos space straight x over denominator straight x end fraction minus sin space straight x space log space straight x close square brackets therefore space du over dx equals space straight x to the power of cos space straight x end exponent open square brackets fraction numerator cos space straight x over denominator straight x end fraction minus sin space straight x space 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 space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis Also space straight v equals left parenthesis cos space straight x right parenthesis to the power of sin space straight x end exponent therefore space log space straight v equals log left parenthesis cos space straight x right parenthesis to the power of sin space straight x end exponent equals sin space straight x. space log left parenthesis cos space straight x right parenthesis therefore space 1 over straight v dv over dx equals sin space straight x. straight d over dx left square bracket log left parenthesis cos space straight x right parenthesis right square bracket plus log left parenthesis cos space straight x right parenthesis. straight d over dx left parenthesis sin space straight x right parenthesis therefore space dv over dx equals straight v open square brackets sin space straight x. fraction numerator 1 over denominator cos space straight x end fraction. left parenthesis negative sin space straight x right parenthesis plus log left parenthesis cos space straight x right parenthesis cross times cos space straight x close square brackets therefore space dv over dx equals space left parenthesis cos space straight x right parenthesis to the power of sin space straight x end exponent left square bracket negative sin space straight x. tan space straight x plus cos space straight x. log left parenthesis cos space straight x right parenthesis right square bracket 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 3 right parenthesis
From (1), (2),(3), we get.
dy over dx equals straight x to the power of cos space straight x end exponent open square brackets fraction numerator cos space straight x over denominator straight x end fraction minus sin space straight x space log space straight x close square brackets plus left parenthesis cos space straight x right parenthesis to the power of sin space straight x end exponent left square bracket negative sin space straight x. tan space straight x plus cos space straight x. log left parenthesis cos space straight x right parenthesis right square bracket

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