Question

Differentiate the following functions w.r.t.x: $log open parentheses square root of fraction numerator 1 minus cos space straight x over denominator 1 plus cos space straight x end fraction end root close parentheses$

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Solution

Let space space space space space straight y equals log open parentheses square root of fraction numerator 1 minus cos space straight x over denominator 1 plus cos space straight x end fraction end root close parentheses equals log open parentheses square root of fraction numerator 2 sin squared 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 end root close parentheses equals log open parentheses tan straight x over 2 close parentheses therefore space dy over dx equals fraction numerator 1 over denominator tan straight x over 2 end fraction. sec squared straight x over 2.1 half equals fraction numerator cos begin display style straight x over 2 end style over denominator sin begin display style straight x over 2 end style end fraction. fraction numerator 1 over denominator cos squared begin display style straight x over 2 end style end fraction.1 half equals fraction numerator 1 over denominator 2 sin begin display style straight x over 2 end style cos begin display style straight x over 2 end style end fraction space space space space space space space space space space space space space equals fraction numerator 1 over denominator sin space straight x end fraction equals cosec space straight x

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