Question

Differentiate the following functions w.r.t. x : $left parenthesis straight x space cos space straight x right parenthesis to the power of straight x plus left parenthesis straight x space sin space straight x right parenthesis to the power of 1 over straight x end exponent$

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Solution

Let space space space space straight y equals left parenthesis straight x space cos space straight x right parenthesis to the power of straight x plus left parenthesis straight x space sin space straight x right parenthesis to the power of 1 over straight x end exponent Put space left parenthesis straight x space cos space straight x right parenthesis to the power of straight x equals straight u comma space left parenthesis straight x space sin space straight x right parenthesis to the power of 1 over 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... left parenthesis 1 right parenthesis Now space straight u equals left parenthesis straight x space cos space straight x right parenthesis to the power of straight x space space space space space space rightwards double arrow space log space straight u equals log left parenthesis straight x space cos space straight x right parenthesis to the power of straight x therefore space log space straight u equals straight x. log left parenthesis straight x space cos space straight x right parenthesis therefore 1 over straight u du over dx equals straight x. fraction numerator straight x left parenthesis negative sin space straight x right parenthesis plus cos space straight x.1 over denominator straight x space cos space straight x end fraction plus log left parenthesis straight x space cos space straight x right parenthesis.1 therefore space du over dx equals left parenthesis straight x space cos space straight x right parenthesis to the power of straight x open square brackets fraction numerator cos space straight x minus straight x space sin space straight x over denominator cos space straight x end fraction plus log left parenthesis straight x space cos space straight x right parenthesis 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... left parenthesis 2 right parenthesis Also space straight v equals left parenthesis straight x space sin space straight x right parenthesis to the power of 1 over straight x end exponent space space space space rightwards double arrow space log space straight v equals log left parenthesis straight x space sin space straight x right parenthesis to the power of 1 over straight x end exponent therefore space space space space space space log space straight v equals 1 over straight x. log left parenthesis straight x space sin space straight x right parenthesis therefore space 1 over straight v dv over dx equals 1 over straight x. fraction numerator straight x space cos space straight x plus sin space straight x.1 over denominator straight x space sin space straight x end fraction plus log left parenthesis straight x space sin space straight x right parenthesis. open parentheses negative 1 over straight x squared close parentheses space space space space space space space space space space space space space space space space space space space equals fraction numerator straight x space cos space straight x plus sin space straight x over denominator straight x squared space sin space straight x end fraction minus fraction numerator log left parenthesis straight x space sin space straight x right parenthesis over denominator straight x squared end fraction therefore 1 over straight v dv over dx equals fraction numerator straight x space cos space straight x plus sin space straight x minus sin space straight x. log left parenthesis straight x space sin space straight x right parenthesis over denominator straight x squared space sin space straight x end fraction therefore space dv over dx equals left parenthesis straight x space sin space straight x right parenthesis to the power of 1 over straight x end exponent open square brackets fraction numerator straight x space cos space straight x plus sin space straight x minus sin space straight x. log left parenthesis straight x space sin space straight x right parenthesis over denominator straight x squared space sin space straight x end fraction 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... 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 comma dy over dx equals left parenthesis straight x space cos space straight x right parenthesis to the power of straight x open square brackets fraction numerator cos space straight x minus straight x space sin space straight x over denominator cos space straight x end fraction plus log left parenthesis straight x space cos space straight x right parenthesis close square brackets plus left parenthesis straight x space sin space straight x right parenthesis to the power of 1 over straight x end exponent open square brackets fraction numerator straight x space cos space straight x plus sin space straight x minus sin space straight x. log left parenthesis straight x space sin space straight x right parenthesis over denominator straight x squared space sin space straight x end fraction close square brackets

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Continuity And Differentiability

Differentiate the following functions w.r.t. x : $left parenthesis straight x space cos space straight x right parenthesis to the power of straight x plus left parenthesis straight x space sin space straight x right parenthesis to the power of 1 over straight x end exponent$

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