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Continuity And Differentiability

Question

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Differentiate :
$log square root of fraction numerator 1 plus sin space straight x over denominator 1 minus sin space straight x end fraction end root$ w.r.t.x.

Solution

Let space space space space space straight y equals log square root of fraction numerator 1 plus sin space straight x over denominator 1 minus sin space straight x end fraction end root therefore space space space space space space space straight y equals log open parentheses fraction numerator 1 plus sin space straight x over denominator 1 minus sin space straight x end fraction close parentheses to the power of 1 half end exponent equals 1 half log open parentheses fraction numerator 1 plus sin space straight x over denominator 1 minus sin space straight x end fraction close parentheses space space space space space space space space space space space space space equals 1 half left square bracket log left parenthesis 1 plus sin space straight x right parenthesis minus log left parenthesis 1 minus sin space straight x right parenthesis right square bracket therefore space dy over dx equals 1 half open square brackets fraction numerator 1 over denominator 1 plus sin space straight x end fraction plus fraction numerator 1 over denominator 1 minus sin space straight x end fraction close square brackets equals 1 half cos space straight x open square brackets fraction numerator 1 minus sin space straight x plus 1 plus sin space straight x over denominator left parenthesis 1 plus sin space straight x right parenthesis left parenthesis 1 minus sin space straight x right parenthesis end fraction close square brackets space space space space space space space space space space space space space equals 1 half cos space straight x open square brackets fraction numerator 2 over denominator 1 minus sin squared straight x end fraction close square brackets equals 1 half cos space straight x open square brackets fraction numerator 2 over denominator cos squared straight x end fraction close square brackets equals fraction numerator 1 over denominator cos space straight x end fraction equals sec space straight x.

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