Question

$If space straight y equals left parenthesis tan space straight x right parenthesis to the power of tan space straight x to the power of tan space straight x.... infinity end exponent end exponent comma space prove space that space dy over dx equals 2 space at space straight x equals straight pi over 2$

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Solution

Here space straight y equals left parenthesis tan space straight x right parenthesis to the power of tan space straight x to the power of tan space straight x.... infinity end exponent end exponent therefore space straight y equals left parenthesis tan space straight x right parenthesis to the power of straight y therefore space log space straight y equals log left parenthesis tan space straight x right parenthesis to the power of straight y space rightwards double arrow space log space straight y equals straight y. log space tan space straight x Differentiating space straight w. straight r. straight t comma space we space get comma space space space space space 1 over straight y dy over dx equals straight y. fraction numerator sec squared straight x over denominator tan space straight x end fraction plus log space tan space straight x dy over dx therefore space open parentheses 1 over straight y minus log space tan space straight x close parentheses dy over dx equals fraction numerator straight y space sec squared straight x over denominator tan space straight x end fraction therefore space open parentheses fraction numerator 1 minus log space tan space straight x over denominator straight y end fraction close parentheses dy over dx equals fraction numerator straight y space sec squared straight x over denominator tan space straight x end fraction therefore space space space space space dy over dx equals negative fraction numerator straight y squared sec squared straight x over denominator tan space straight x open parentheses 1 minus straight y space log space tan space straight x close parentheses end fraction When space straight x equals straight pi over 2 comma space dy over dx equals fraction numerator left parenthesis 1 right parenthesis sec squared begin display style straight pi over 4 end style over denominator tan begin display style straight pi over 4 end style open parentheses 1 minus 1 space log space tan begin display style straight pi over 4 end style close parentheses end fraction space space space space space space space space space space space space space space space space open square brackets because space straight y equals 1 space when space straight x equals straight pi over 4 close square brackets 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 equals fraction numerator open parentheses square root of 2 close parentheses squared over denominator left parenthesis 1 right parenthesis left parenthesis 1 minus log space 1 right parenthesis end fraction equals 2

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