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Continuity And Differentiability

Question
CBSEENMA12035091

Differentiate the following functions w.r.t. x :straight x to the power of straight x to the power of straight x end exponent plus straight x to the power of sin space straight x end exponent

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

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