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Continuity And Differentiability

Question
CBSEENMA12035376
Wired Faculty App

If y = sin (sin x), prove that
fraction numerator straight d squared straight y over denominator dx squared end fraction plus tan space straight x space dy over dx plus space straight y space cos squared space straight x equals 0

Solution
Here space space straight y space equals space sin space left parenthesis sin space straight x right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 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 therefore space dy over dx equals cos space left parenthesis sin space straight x right parenthesis. cos space straight x space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 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 differentiating space both space sides space straight w. straight r. straight t. straight x comma space space space space space dy over dx equals cos left parenthesis sin space straight x right parenthesis. straight d over dx left parenthesis cos space straight x right parenthesis plus cos space straight x. straight d over dx left square bracket cos left parenthesis sin space straight x right parenthesis right square bracket space space space space space space space space space space space space space equals cos left parenthesis sin space straight x right parenthesis. left parenthesis negative sin space straight x right parenthesis plus cos space straight x. left square bracket negative sin left parenthesis sin space straight x right parenthesis. cos space straight x right square bracket space space space space space space space space space space space space space equals negative sin space straight x. space cos left parenthesis sin space straight x right parenthesis minus cos 2 space straight x. space sin left parenthesis sin space straight x right parenthesis space space space space space space space space space space space space space equals negative fraction numerator sin space straight x over denominator cos space straight x end fraction. left square bracket cos left parenthesis sin space straight x right parenthesis. space cos space straight x right square bracket minus cos 2 space straight x. sin left parenthesis sin space straight x right parenthesis space space space space space space space space space space space space space equals negative tan space straight x. dy over dx minus cos squared space straight x. straight y space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space 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 square bracket because space of space left parenthesis 1 right parenthesis comma left parenthesis 2 right parenthesis right square bracket therefore space fraction numerator straight d squared straight y over denominator dx squared end fraction plus tan space straight x space dy over dx plus straight y space cos squared space straight x equals 0.

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