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Continuity And Differentiability

Question

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

Solution

Let space space space space space space space space straight y equals straight e to the power of sin space straight x end exponent plus left parenthesis tan space straight x right parenthesis to the power of straight x Put space straight e to the power of sin space straight x end exponent equals straight u comma space left parenthesis tan space straight x right parenthesis to the power of straight x equals straight u therefore space space space space space space space straight y equals straight u plus straight v therefore space dy over dx equals du over dx plus dv over dx Now space straight u equals straight e to the power of sin space straight x end exponent therefore space du over dx equals straight e to the power of sin space straight x end exponent. cos space straight x Also space space space space space space space straight v equals left parenthesis tan space straight x right parenthesis to the power of straight x therefore space space space space space log space straight v equals log left parenthesis tan space straight x right parenthesis to the power of straight x therefore space space space space space log space straight v equals straight x. log left parenthesis tan space straight x right parenthesis therefore space 1 over straight v dv over dx equals straight x. fraction numerator sec squared straight x over denominator tan space straight x end fraction plus log space tan space straight x.1 therefore space space space space space space space dv over dx equals straight v open square brackets straight x. fraction numerator 1 over denominator cos squared straight x end fraction. fraction numerator cos space straight x over denominator sin space straight x end fraction plus log space tan space straight x close square brackets therefore space space space space space space space dv over dx equals left parenthesis tan space straight x right parenthesis to the power of straight x open square brackets fraction numerator straight x over denominator sin space xcos space straight x end fraction plus log space tan space straight x close square brackets 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 space space space space space space space space space space space dy over dx equals straight e to the power of sin space straight x end exponent. cos space straight x plus left parenthesis tan space straight x right parenthesis to the power of straight x open square brackets fraction numerator straight x over denominator sin space xcos space straight x end fraction plus log space tan space straight x close square brackets

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