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Continuity And Differentiability

Question
CBSEENMA12034831
Wired Faculty App

If space straight y equals straight x to the power of straight x to the power of straight x........ infinity end exponent end exponent space prove space that space straight x dy over dx equals fraction numerator straight y squared over denominator 1 minus straight y space log space straight x end fraction

Solution
because space straight y equals straight x to the power of straight x to the power of straight x........ infinity end exponent end exponent therefore space straight y equals straight x space space space space space space space space space space space rightwards double arrow space space log space straight y equals log space straight x to the power of straight x space space space space space rightwards double arrow space space space log space straight y equals straight y space log space straight x Differentiate space both space sides space straight w. straight r. straight t. straight x 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 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 rightwards double arrow 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 space space space space space space space rightwards double arrow space straight x dy over dx equals fraction numerator straight y squared over denominator 1 minus straight y space log space straight x end fraction space

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