Question

Differentiate the following functions w.r.t. x : $straight x to the power of tan space straight x end exponent plus left parenthesis sin space straight x right parenthesis to the power of cos space straight x end exponent$

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Solution

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

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Continuity And Differentiability

Differentiate the following functions w.r.t. x : $straight x to the power of tan space straight x end exponent plus left parenthesis sin space straight x right parenthesis to the power of cos space straight x end exponent$

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