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Continuity And Differentiability

Question

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Differentiate the following w.r.t.x: $fraction numerator sin squared straight x over denominator 1 plus cos squared space straight x end fraction$

Solution

Let space space space space space straight y equals fraction numerator sin squared straight x over denominator 1 plus cos squared space straight x end fraction therefore space dy over dx equals fraction numerator left parenthesis 1 plus cos squared space straight x right parenthesis. begin display style straight d over dx end style left parenthesis sin squared straight x right parenthesis minus sin squared straight x. begin display style straight d over dx end style left parenthesis 1 plus cos squared space straight x right parenthesis over denominator left parenthesis 1 plus cos squared space straight x right parenthesis squared end fraction space space space space space space space space space space space space space equals fraction numerator left parenthesis 1 plus cos squared space straight x right parenthesis. left parenthesis 2 sin space straight x space cos space straight x right parenthesis minus sin squared straight x. left parenthesis negative 2 cos space straight x space sin space straight x right parenthesis over denominator left parenthesis 1 plus cos squared space straight x right parenthesis squared end fraction space space space space space space space space space space space space space equals fraction numerator 2 sin space xcos space straight x left square bracket 1 plus cos squared space straight x plus sin squared straight x right square bracket over denominator left parenthesis 1 plus cos squared space straight x right parenthesis squared end fraction equals fraction numerator sin space 2 straight x left square bracket 1 plus 1 right square bracket over denominator left parenthesis 1 plus cos squared space straight x right parenthesis squared end fraction equals fraction numerator 2 sin space 2 straight x over denominator left parenthesis 1 plus cos squared space straight x right parenthesis squared end fraction

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