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Continuity And Differentiability

Question

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Differentiate the following w.r.t.x: $square root of fraction numerator straight x squared minus 2 ax over denominator straight a squared minus 2 ab end fraction end root$

Solution

Let space space space space space space space straight y equals square root of fraction numerator straight x squared minus 2 ax over denominator straight a squared minus 2 ab end fraction end root equals square root of fraction numerator straight x squared minus 2 ax over denominator straight a squared minus 2 ab end fraction end root equals fraction numerator 1 over denominator square root of straight a squared minus 2 ab end root end fraction left parenthesis straight x squared minus 2 ax right parenthesis to the power of begin inline style bevelled 1 half end style end exponent therefore space dy over dx equals fraction numerator 1 over denominator square root of straight a squared minus 2 ab end root end fraction.1 half left parenthesis straight x squared minus 2 ax right parenthesis to the power of negative 1 half end exponent. straight d over dx left parenthesis straight x squared minus 2 ax right parenthesis space space space space space space space space space space space space space equals fraction numerator 1 over denominator square root of straight a squared minus 2 ab end root end fraction. fraction numerator 1 over denominator square root of straight x squared minus 2 ax end root end fraction. left parenthesis 2 straight x minus 2 straight a right parenthesis equals fraction numerator straight x minus straight a over denominator square root of straight a squared minus 2 ab end root square root of straight x squared minus 2 ax end root end fraction

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