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Continuity And Differentiability

Question

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Differentiate : $straight x square root of straight x squared plus 1 end root plus log open parentheses square root of straight x squared plus 1 end root close parentheses$ w.r.t x.

Solution

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

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