Question

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

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Solution

Let space straight y equals log open parentheses fraction numerator square root of straight a plus straight x end root plus square root of straight a minus straight x end root over denominator square root of straight a plus straight x end root minus square root of straight a minus straight x end root end fraction close parentheses space space space space space space equals log open square brackets fraction numerator square root of straight a plus straight x end root plus square root of straight a minus straight x end root over denominator square root of straight a plus straight x end root minus square root of straight a minus straight x end root end fraction cross times fraction numerator square root of straight a plus straight x end root plus square root of straight a minus straight x end root over denominator square root of straight a plus straight x end root plus square root of straight a minus straight x end root end fraction close square brackets space space space space space space equals log open square brackets fraction numerator straight a plus straight x plus straight a minus straight x plus 2 square root of straight a squared minus straight x squared end root over denominator straight a plus straight x minus straight x left parenthesis straight a minus straight x right parenthesis end fraction close square brackets equals log open parentheses fraction numerator straight a plus square root of straight a squared minus straight x squared end root over denominator straight x end fraction close parentheses space space space space space space equals log open parentheses straight a plus square root of straight a squared minus straight x squared end root close parentheses minus log space straight x therefore dy over dx equals fraction numerator 1 over denominator straight a plus square root of straight a squared minus straight x squared end root end fraction cross times straight d over dx left square bracket straight a plus left parenthesis straight a squared minus straight x squared right parenthesis right square bracket to the power of begin inline style 1 half end style end exponent minus 1 over straight x space space space space space space equals fraction numerator 1 over denominator straight a plus square root of straight a squared minus straight x squared end root end fraction cross times 1 half left parenthesis straight a squared minus straight x squared right parenthesis to the power of begin inline style 1 half end style minus 1 end exponent cross times straight d over dx left parenthesis straight a squared minus straight x squared right parenthesis minus 1 over straight x space space space space space space equals fraction numerator 1 over denominator straight a plus square root of straight a squared minus straight x squared end root end fraction cross times fraction numerator 1 over denominator 2 square root of straight a squared minus straight x squared end root end fraction left parenthesis negative 2 straight x right parenthesis minus 1 over straight x space space space space space space equals fraction numerator negative straight x over denominator square root of straight a squared minus straight x squared end root. open square brackets straight a plus square root of straight a squared minus straight x squared end root close square brackets end fraction minus 1 over straight x equals negative fraction numerator straight x squared plus straight a square root of straight a squared minus straight x squared end root plus straight a squared minus straight x squared over denominator straight x square root of straight a squared minus straight x squared end root open square brackets straight a plus square root of straight a squared minus straight x squared end root close square brackets end fraction space space space space space space equals fraction numerator straight a open square brackets straight a plus square root of straight a squared minus straight x squared end root close square brackets over denominator straight x square root of straight a squared minus straight x squared end root open square brackets straight a plus square root of straight a squared minus straight x squared end root close square brackets end fraction equals negative fraction numerator straight a over denominator straight x square root of straight a squared minus straight x squared end root end fraction

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