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Continuity And Differentiability

Question
CBSEENMA12035078
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Differentiate the following functions w.r.t. x : straight x to the power of straight x space cos space straight x end exponent plus fraction numerator straight x squared plus 1 over denominator straight x squared minus 1 end fraction

Solution
Let space space space space space space space space space space space space space straight y equals straight x to the power of straight x space cos space straight x end exponent plus fraction numerator straight x squared plus 1 over denominator straight x squared minus 1 end fraction Put space space straight x to the power of straight x space cos space straight x end exponent equals straight u comma space fraction numerator straight x squared plus 1 over denominator straight x squared minus 1 end fraction equals straight v therefore space space space space space space space space space space space space space space straight y equals straight u plus straight v space rightwards double arrow 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... left parenthesis 1 right parenthesis Now space space space space space space space space straight u equals straight x to the power of straight x space cos space straight x end exponent therefore space space space space space space log space straight u equals log left parenthesis straight x to the power of straight x space cos space straight x end exponent right parenthesis space space rightwards double arrow space log space straight u equals straight x space cos space straight x. log space straight x therefore space 1 over straight u du over dx equals straight x space cos space straight x.1 over straight x plus cos space xlog space straight x.1 plus straight x space log space straight x. left parenthesis negative sin space straight x right parenthesis therefore space space space space space space space du over dx equals straight u left square bracket cos space straight x plus cos space straight x space log space straight x minus straight x space sin space xlog space straight x right square bracket therefore space space space space space space space du over dx equals space straight x to the power of straight x space cos space straight x end exponent left square bracket cos space straight x plus cos space straight x space log space straight x minus straight x space sin space xlog space straight x 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... left parenthesis 2 right parenthesis Again space space space space space space straight v equals fraction numerator straight x squared plus 1 over denominator straight x squared minus 1 end fraction therefore space space space space space space space dv over dx equals fraction numerator left parenthesis straight x squared minus 1 right parenthesis.2 straight x minus left parenthesis straight x squared plus 1 right parenthesis.2 straight x over denominator left parenthesis straight x squared minus 1 right parenthesis squared end fraction equals fraction numerator 2 straight x cubed minus 2 straight x minus 2 straight x cubed minus 2 straight x over denominator left parenthesis straight x squared minus 1 right parenthesis squared end fraction therefore space space space space space space space dv over dx equals fraction numerator 4 straight x over denominator left parenthesis straight x squared minus 1 right parenthesis squared end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 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 straight x space cos space straight x end exponent left square bracket cos space straight x plus cos space straight x space log space straight x minus straight x space sin space xlog space straight x right square bracket plus fraction numerator 4 straight x over denominator left parenthesis straight x squared minus 1 right parenthesis squared end fraction

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