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Continuity And Differentiability

Question

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

Solution

Let space space space space space straight y equals fraction numerator sin space straight x over denominator 1 plus cos space straight x end fraction equals fraction numerator 2 sin begin display style straight x over 2 end style cos begin display style straight x over 2 end style over denominator 2 cos squared begin display style straight x over 2 end style end fraction equals fraction numerator sin begin display style straight x over 2 end style over denominator cos begin display style straight x over 2 end style end fraction equals tan straight x over 2 therefore space dy over dx equals sec squared straight x over 2. straight d over dx open parentheses straight x over 2 close parentheses equals sec squared straight x over 2 cross times 1 half equals 1 half sec squared straight x over 2

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