Question

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

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Solution

Let space space space space straight y equals left parenthesis sin space straight x right parenthesis to the power of tan space space straight x end exponent plus left parenthesis cos space straight x right parenthesis to the power of sec space straight x end exponent Put space left parenthesis sin space straight x right parenthesis to the power of tan space space straight x end exponent equals straight u comma space left parenthesis cos space straight x right parenthesis to the power of sec space straight x end exponent equals straight v therefore 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 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 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 straight u equals left parenthesis sin space straight x right parenthesis to the power of tan space space straight x end exponent therefore space log space straight u equals log left parenthesis sin space straight x right parenthesis to the power of tan space space straight x end exponent space space rightwards double arrow space log space straight u equals tan space straight x. log left parenthesis sin space straight x right parenthesis therefore space 1 over straight u du over dx equals tan space straight x. fraction numerator 1 over denominator sin space straight x end fraction. cos space straight x plus sec squared straight x. space log left parenthesis sin space straight x right parenthesis therefore space space space space space space space du over dx equals straight u left square bracket tan space straight x. cot space straight x plus sec squared straight x. log left parenthesis sin space straight x right parenthesis right square bracket therefore space space space space space space space du over dx equals left parenthesis sin space straight x right parenthesis to the power of tan space space straight x end exponent left square bracket 1 plus sec squared straight x. space log left parenthesis sin 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 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 space space space straight v equals left parenthesis cos space straight x right parenthesis to the power of sec 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 sec space straight x end exponent therefore space log space straight v equals sec. log left parenthesis cos space straight x right parenthesis therefore space 1 over straight v dv over dx equals sec 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. sec space xtan space straight x therefore space space space space space space space dv over dx equals straight v left square bracket negative sec space straight x space tan space straight x plus sec space straight x space tan space straight x. log left parenthesis cos space straight x right parenthesis right square bracket therefore space space space space space space space dv over dx equals left parenthesis cos space straight x right parenthesis to the power of sec space straight x end exponent left square bracket negative sec space straight x. tan space straight x plus sec space straight x space tan 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... 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 sin space straight x right parenthesis to the power of tan space space straight x end exponent left square bracket 1 plus sec squared straight x. space log left parenthesis sin space straight x right parenthesis right square bracket plus cos space straight x right parenthesis to the power of sec space straight x end exponent left square bracket negative sec space straight x. tan space straight x plus sec space straight x space tan space straight x. log left parenthesis cos space straight x right parenthesis right square bracket

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Continuity And Differentiability

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

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