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Continuity And Differentiability

Question
CBSEENMA12035061
Wired Faculty App

If space straight y equals straight x to the power of cot space straight x end exponent plus left parenthesis sin space straight x right parenthesis to the power of straight x comma space find space dy over dx.

Solution
Here space space space space space straight y equals straight x to the power of cot space straight x end exponent plus left parenthesis sin space straight x right parenthesis to the power of straight x Put space straight x to the power of cot space straight x end exponent equals straight u comma space left parenthesis sin space straight x right parenthesis to the power of straight x 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... left parenthesis 1 right parenthesis Now space straight u equals straight x to the power of cot space straight x end exponent therefore space log space straight u equals log space straight x to the power of cot space straight x end exponent therefore space log space straight u equals cot space straight x. space log space straight x therefore space 1 over straight u du over dx equals left parenthesis cot space straight x right parenthesis. open parentheses 1 over straight x close parentheses plus left parenthesis log space straight x right parenthesis. left parenthesis negative coesc squared straight x right parenthesis therefore space du over dx equals straight u open square brackets fraction numerator cot space straight x over denominator straight x end fraction minus log space straight x. coesc squared straight x close square brackets therefore space du over dx equals straight x to the power of cot space straight x end exponent open square brackets fraction numerator cot space straight x over denominator straight x end fraction minus log space straight x. coesc squared straight x 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 space space space space space... left parenthesis 2 right parenthesis
Again space straight v equals left parenthesis sin space straight x right parenthesis to the power of straight x therefore space log space straight v equals log left parenthesis sin right parenthesis to the power of straight x therefore space log space straight v equals straight x. log left parenthesis sin space straight x right parenthesis therefore space 1 over straight v dv over dx equals straight x. open parentheses fraction numerator 1 over denominator sin space straight x end fraction. cos space straight x close parentheses plus log left parenthesis sin space straight x right parenthesis.1 therefore space dv over dx equals straight v left square bracket straight x space cot space straight x plus log left parenthesis sin space straight x right parenthesis right square bracket therefore space dv over dx equals left parenthesis sin space straight x right parenthesis to the power of straight x left square bracket straight x space cot space straight x plus 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 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 straight x to the power of cot space straight x end exponent open square brackets fraction numerator cot space straight x over denominator straight x end fraction minus log space straight x. cosec squared straight x close square brackets plus left parenthesis sin space straight x right parenthesis to the power of straight x left square bracket straight x space cot space straight x plus log left parenthesis sin space straight x right parenthesis right square bracket

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