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Continuity And Differentiability

Question
CBSEENMA12035370
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Find space fraction numerator straight d squared straight y over denominator dx squared end fraction comma space if space straight y equals straight e to the power of straight x space sin end exponent space to the power of straight x.

Solution
space space space space space space space space space space space space space straight y equals straight e to the power of straight x space sin end exponent space to the power of straight x therefore space space space dy over dx equals straight e to the power of straight x space sin end exponent space to the power of straight x. straight d over dx left parenthesis straight x space sin space straight x right parenthesis rightwards double arrow space space dy over dx equals straight e to the power of straight x space sin end exponent space to the power of straight x. left parenthesis straight x space cos space straight x plus sin space straight x right parenthesis therefore space fraction numerator straight d squared straight y over denominator dx squared end fraction equals straight e to the power of straight x space sin end exponent space to the power of straight x. left parenthesis negative straight x space sin space straight x plus cos space straight x plus cos space straight x right parenthesis plus left parenthesis straight x space cos space straight x plus sin space straight x right parenthesis. straight e to the power of straight x space sin end exponent space to the power of straight x. left parenthesis straight x space cos space straight x plus sin space straight x right parenthesis therefore space fraction numerator straight d squared straight y over denominator dx squared end fraction equals straight e to the power of straight x space sin end exponent space to the power of straight x left square bracket negative straight x space sin space straight x plus 2 space cos space straight x plus left parenthesis straight x space cos space straight x plus sin space straight x right parenthesis squared right square bracket

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