Question

$If space straight y equals straight x plus fraction numerator 1 over denominator straight x plus begin display style fraction numerator 1 over denominator straight x plus... end fraction end style end fraction comma space prove space that space dy over dx equals fraction numerator straight y over denominator 2 straight y minus straight x end fraction.$

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Solution

Here space straight y equals straight x plus fraction numerator 1 over denominator straight x plus begin display style fraction numerator 1 over denominator straight x plus... end fraction end style end fraction therefore space space space space space space straight y equals straight x plus 1 over straight y therefore space space space space straight y squared equals straight x space straight y plus 1 Differentiating space both space sides space straight w. straight r. straight t. space straight x comma space we space get comma 2 straight y dy over dx equals straight x dy over dx plus straight y.1 plus 0 therefore space left parenthesis 2 straight y minus straight x right parenthesis dy over dx equals straight y therefore space dy over dx equals fraction numerator straight y over denominator 2 straight y minus straight x end fraction

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