Question

Differentiate the following w.r.t.x: $open parentheses fraction numerator straight x plus square root of straight x squared minus straight a squared end root over denominator straight x minus square root of straight x squared minus straight a squared end root end fraction close parentheses$

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Solution

Let space straight y equals open parentheses fraction numerator straight x plus square root of straight x squared minus straight a squared end root over denominator straight x minus square root of straight x squared minus straight a squared end root end fraction close parentheses space space space space space space space space space equals log open parentheses fraction numerator straight x plus square root of straight x squared minus straight a squared end root over denominator straight x minus square root of straight x squared minus straight a squared end root end fraction cross times fraction numerator straight x plus square root of straight x squared minus straight a squared end root over denominator straight x plus square root of straight x squared minus straight a squared end root end fraction close parentheses space space space space space space space space space equals log open square brackets fraction numerator open parentheses straight x plus square root of straight x squared minus straight a squared end root close parentheses squared over denominator straight x squared minus left parenthesis straight x squared minus straight a squared right parenthesis end fraction close square brackets equals log open square brackets open parentheses straight x plus square root of straight x squared minus straight a squared end root close parentheses squared over straight a squared close square brackets space space space space space space space space space equals log open parentheses straight x plus square root of straight x squared minus straight a squared end root close parentheses squared minus log space straight a squared therefore space straight y equals 2 space log open parentheses straight x plus square root of straight x squared minus straight a squared end root close parentheses minus 2 space log space straight a therefore space dy over dx equals fraction numerator 2 over denominator straight x plus square root of straight x squared minus straight a squared end root end fraction cross times open square brackets 1 plus fraction numerator 2 over denominator 2 square root of straight x squared minus straight a squared end root end fraction close square brackets minus 0 space space space space space space space space space space space space space equals fraction numerator 2 over denominator straight x plus square root of straight x squared minus straight a squared end root end fraction cross times fraction numerator square root of straight x squared minus straight a squared end root plus straight x over denominator square root of straight x squared minus straight a squared end root end fraction therefore space dy over dx equals fraction numerator 2 over denominator square root of straight x squared minus straight a squared end root end fraction

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Continuity And Differentiability

Differentiate the following w.r.t.x: $open parentheses fraction numerator straight x plus square root of straight x squared minus straight a squared end root over denominator straight x minus square root of straight x squared minus straight a squared end root end fraction close parentheses$

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