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Continuity And Differentiability

Question

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Differentiate $fraction numerator 1 over denominator straight x plus square root of 1 plus straight x squared end root end fraction$ w.r.t x.

Solution

Let space straight y equals fraction numerator 1 over denominator straight x plus square root of 1 plus straight x squared end root end fraction equals fraction numerator 1 over denominator straight x plus square root of 1 plus straight x squared end root end fraction cross times fraction numerator straight x minus square root of 1 plus straight x squared end root over denominator straight x minus square root of 1 plus straight x squared end root end fraction equals fraction numerator straight x minus square root of 1 plus straight x squared end root over denominator straight x squared minus left parenthesis 1 plus straight x squared right parenthesis end fraction space space space space space space space space equals fraction numerator straight x plus square root of 1 plus straight x squared end root over denominator negative 1 end fraction equals negative straight x plus square root of 1 plus straight x squared end root therefore space dy over dx equals negative 1 plus fraction numerator 2 straight x over denominator 2 square root of 1 plus straight x squared end root end fraction equals negative 1 plus fraction numerator straight x over denominator square root of 1 plus straight x squared end root end fraction

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