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Continuity And Differentiability

Question
CBSEENMA12035047
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Differentiate xsin x+(sin x)x w.r.t x.

Solution
Let 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 straight x 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 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 space space space space... left parenthesis 1 right parenthesis Now space straight u equals straight x to the power of sin space straight x end exponent therefore space log space straight u equals log space straight x to the power of sin space straight x end exponent rightwards double arrow space log space straight u equals sin space straight x. log space straight x therefore space 1 over straight u du over dx equals sin space straight x.1 over straight x plus log space straight x. space cos space straight x therefore space du over dx equals straight u open parentheses straight x over straight x plus cos space straight x. log space straight x close parentheses therefore space du over dx equals straight x to the power of sin space straight x end exponent open parentheses straight x over straight x plus cos space straight x. log space straight x close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 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 sin space straight x right parenthesis to the power of straight x therefore space log space straight v equals log left parenthesis sin space straight x right parenthesis to the power of straight x rightwards double arrow space log space straight v equals straight x. log left parenthesis sin space straight x right parenthesis rightwards double arrow space 1 over straight v dv over dx equals straight x. fraction numerator cos space straight x over denominator sin space straight x end fraction plus left parenthesis log space 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 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 sin space straight x end exponent open parentheses straight x over straight x plus cos space straight x. log space straight x close parentheses 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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