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Continuity And Differentiability

Question

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$If space equals log square root of fraction numerator 1 plus sin squared straight x over denominator 1 minus tan space straight x end fraction end root comma space find space dy over dx.$

Solution

Here space straight y equals log square root of fraction numerator 1 plus sin squared straight x over denominator 1 minus tan space straight x end fraction end root therefore space space space space space space straight y equals 1 half log open parentheses fraction numerator 1 plus sin squared straight x over denominator 1 minus tan space straight x end fraction close parentheses rightwards double arrow space space space space space space straight y equals 1 half left square bracket log left parenthesis 1 plus sin squared straight x right parenthesis minus log left parenthesis 1 minus tan space straight x right parenthesis right square bracket therefore space dy over dx equals 1 half open square brackets fraction numerator 2 sin space straight x space cos space straight x over denominator 1 plus sin squared straight x end fraction minus fraction numerator negative sec squared straight x over denominator 1 minus tan space straight x end fraction close square brackets therefore space dy over dx equals 1 half open square brackets fraction numerator sin space 2 straight x over denominator 1 plus sin squared straight x end fraction plus fraction numerator sec squared straight x over denominator 1 minus tan space straight x end fraction close square brackets

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