Question

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

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Solution

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

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