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Continuity And Differentiability

Question

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Use Chain Rule to differentiate w.r.t.x: sin (cos x²)

Solution

space space space space space space space space space space space space space space space space space space straight f left parenthesis straight x right parenthesis equals sin left parenthesis cos left parenthesis straight x squared right parenthesis right parenthesis The space fuction space straight f left parenthesis straight x right parenthesis space is space straight a space composition space straight f left parenthesis straight x right parenthesis equals left parenthesis wovou right parenthesis left parenthesis straight x right parenthesis space of space the space three space function space straight u comma straight v space and space straight w comma space where straight u left parenthesis straight x right parenthesis equals straight x to the power of 2 comma space end exponent straight v left parenthesis straight t right parenthesis equals cos space straight t space and space straight w left parenthesis straight s right parenthesis equals sin space straight s Put space straight t equals straight u left parenthesis straight x right parenthesis equals straight x squared comma space straight s equals straight v left parenthesis straight t right parenthesis equals cos space straight t dw over ds equals cos space straight s comma space ds over dt equals negative sin space straight t comma space dt over dx equals 2 straight x By space Chain space Rule comma space we space have comma space space space space space space space space space space space space space space space space space space space space space space df over dx equals dw over ds. ds over dt. dt over dx equals left parenthesis cos space straight s right parenthesis. left parenthesis negative sin space straight t right parenthesis. left parenthesis 2 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 equals cos space left parenthesis cos space straight t right parenthesis. left parenthesis negative sin space straight t right parenthesis. left parenthesis 2 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 equals cos left parenthesis cos left parenthesis straight x squared right parenthesis right parenthesis. left parenthesis negative sin left parenthesis straight x squared right parenthesis right parenthesis. left parenthesis 2 straight x right parenthesis therefore space straight d over dx left square bracket sin left parenthesis cos left parenthesis straight x squared right parenthesis right parenthesis right square bracket equals negative 2 straight x space sin space straight x squared cos left parenthesis cos space straight x squared right parenthesis

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