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Continuity And Differentiability

Question

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Differentiate the following functions w.r.t.x: $log space tan open parentheses straight pi over 4 plus straight x over 2 close parentheses$

Solution

Let space space space space space straight y equals log space tan open parentheses straight pi over 4 plus straight x over 2 close parentheses therefore space dy over dx equals fraction numerator 1 over denominator open parentheses straight pi over 4 plus straight x over 2 close parentheses end fraction. straight d over dx open square brackets tan open parentheses straight pi over 4 plus straight x over 2 close parentheses close square brackets space space space space space space space space space space space space space equals fraction numerator 1 over denominator tan open parentheses straight pi over 4 plus straight x over 2 close parentheses end fraction. sec squared open parentheses straight pi over 4 plus straight x over 2 close parentheses. straight d over dx open parentheses straight pi over 4 plus straight x over 2 close parentheses space space space space space space space space space space space space space equals fraction numerator cos open parentheses straight pi over 4 plus straight x over 2 close parentheses over denominator sin open parentheses straight pi over 4 plus straight x over 2 close parentheses end fraction. fraction numerator 1 over denominator cos squared open parentheses straight pi over 4 plus straight x over 2 close parentheses end fraction.1 half space space space space space space space space space space space space space equals fraction numerator 1 over denominator 2 sin open parentheses straight pi over 4 plus straight x over 2 close parentheses cos open parentheses straight pi over 4 plus straight x over 2 close parentheses end fraction equals fraction numerator 1 over denominator sin open parentheses straight pi over 2 plus straight x close parentheses end fraction equals fraction numerator 1 over denominator cos space straight x end fraction equals sec space straight x

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