Question

If x^y = e^{x - y}, prove that $dy over dx equals fraction numerator log space straight x over denominator left parenthesis 1 plus log space straight x right parenthesis squared end fraction$

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Solution

straight x to the power of straight y equals straight e to the power of straight x minus straight y end exponent end exponent space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space log space straight x to the power of straight y equals log space straight e to the power of straight x minus straight y end exponent therefore space straight y space log space straight x equals straight x minus straight y space space space space space space space space space space space space space space space space space space space space rightwards double arrow space straight y plus straight y space log space straight x equals straight x therefore space straight y left parenthesis 1 plus log space straight x right parenthesis equals straight x space space space space space space space space space space space space space space space space space space rightwards double arrow space space straight y equals fraction numerator straight x over denominator 1 plus log space straight x end fraction Differentiate space both space sides space straight w. straight r. straight t. straight x comma dy over dx equals fraction numerator left parenthesis 1 plus log space straight x right parenthesis.1 minus straight x. begin display style 1 over straight x end style over denominator left parenthesis 1 plus log space straight x right parenthesis squared end fraction equals fraction numerator 1 plus log space straight x minus 1 over denominator left parenthesis 1 plus log space straight x right parenthesis squared end fraction equals fraction numerator log space straight x over denominator left parenthesis 1 plus log space straight x right parenthesis squared end fraction

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Continuity And Differentiability

If x^y = e^{x - y}, prove that $dy over dx equals fraction numerator log space straight x over denominator left parenthesis 1 plus log space straight x right parenthesis squared end fraction$

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If x^y = e^{x - y}, prove that $dy over dx equals fraction numerator log space straight x over denominator left parenthesis 1 plus log space straight x right parenthesis squared end fraction$