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Continuity And Differentiability

Question

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Differentiate $log open parentheses straight x plus square root of straight a squared plus straight x squared end root close parentheses$ w.r.t x.

Solution

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

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