Question

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

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Solution

Let space space space space space space space space straight y equals straight x to the power of sin 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 sin 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 space rightwards double arrow space space dy over dx equals du over dx plus dv over dx straight u equals straight x to the power of sin space straight x end exponent therefore space log space straight u equals log space sin space straight x. log space straight x therefore space 1 over straight u du over dx equals fraction numerator sin space straight x over denominator straight x end fraction plus cos space straight x. log space straight x therefore space du over dx equals straight x to the power of sin space straight x end exponent open square brackets fraction numerator sin space straight x over denominator straight x end fraction plus cos space straight x. log space straight x close square brackets straight v equals 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 log left parenthesis sin space straight x right parenthesis to the power of cot space straight x end exponent equals cos space straight x. space 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. cot space straight x minus sin space straight x. log space sin space straight x right square bracket 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 sin space straight x end exponent open square brackets fraction numerator sin space straight x over denominator straight x end fraction plus cos space straight x. log space straight x close square brackets 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 : $straight x to the power of sin space straight x end exponent plus left parenthesis sin space straight x right parenthesis to the power of cos space straight x end exponent$

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