Question

$If space straight y equals straight x to the power of straight y comma space prove space that space dy over dx equals fraction numerator straight y squared over denominator straight x left parenthesis 1 minus straight y space log space straight x right parenthesis end fraction.$

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Solution

Here space space space space space straight y equals straight x to the power of straight y therefore space log space straight y equals log space straight x to the power of straight y space space space space space space space space space rightwards double arrow space log space straight y equals straight y space log space straight x Differentiating space both space sides space straight w. straight r. straight t. straight x comma space we space get comma 1 over straight y dy over dx equals straight y.1 over straight x plus log space straight x. dy over dx therefore space open parentheses 1 over straight y minus log space straight x close parentheses dy over dx equals straight y over straight x space space space space rightwards double arrow space open parentheses fraction numerator 1 minus straight y space log space straight x over denominator straight y end fraction close parentheses dy over dx equals straight y over straight x therefore space space space space space space space space space space space space space dy over dx equals fraction numerator straight y squared over denominator straight x left parenthesis 1 minus straight y space log space straight x right parenthesis end fraction.

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