Question

$If space straight x to the power of straight y equals straight y to the power of straight x comma space prove space that space dy over dx equals fraction numerator begin display style straight y over straight x end style minus log space straight y over denominator begin display style straight x over straight y end style minus log space straight x end fraction$

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Solution

space straight x to the power of straight y equals straight y to the power of straight x rightwards double arrow space log space straight x to the power of straight y equals log space straight y to the power of straight x space space space space space space space rightwards double arrow space straight y space log space straight x equals straight x. log space straight y Differentiating space both space sides space straight w. straight r. straight t. straight x comma straight y.1 over straight x plus log space straight x dy over dx equals straight x.1 over straight y. dy over dx plus log space straight y.1 therefore space open parentheses log space straight x minus straight x over straight y close parentheses dy over dx equals log space straight y minus straight y over straight x space space space rightwards double arrow open parentheses straight x over straight y minus log space straight x close parentheses dy over dx equals straight y over straight x minus log space straight y dy over dx equals fraction numerator begin display style straight y over straight x minus log space straight y end style over denominator begin display style straight x over straight y minus log space straight x end style end fraction

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