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Continuity And Differentiability

Question
CBSEENMA12035306
Wired Faculty App

If space left parenthesis tan to the power of negative 1 end exponent straight x right parenthesis plus straight y to the power of cot space straight x end exponent equals 1 comma space find space dy over dx.

Solution
Here space left parenthesis tan to the power of negative 1 end exponent straight x right parenthesis to the power of straight y plus straight y to the power of cot space straight x end exponent equals 1 Put space left parenthesis tan to the power of negative 1 end exponent straight x right parenthesis equals straight u comma space straight y to the power of cot space straight x end exponent equals straight v therefore space straight u plus straight v equals 1 space space space space space space space space space space space space space space space space rightwards double arrow space du over dx plus dv over dx equals 0 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 space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis Now space straight u equals left parenthesis tan to the power of negative 1 end exponent straight x right parenthesis to the power of straight y space rightwards double arrow space log space straight u equals log left parenthesis tan to the power of negative 1 end exponent straight x right parenthesis to the power of straight y space rightwards double arrow space log space straight v equals straight y. log left parenthesis tan to the power of negative 1 end exponent straight x right parenthesis therefore space 1 over straight u du over dx equals straight y. fraction numerator 1 over denominator tan to the power of negative 1 end exponent straight x end fraction. fraction numerator 1 over denominator 1 plus straight x squared end fraction plus dy over dx. log space tan to the power of negative 1 end exponent straight x therefore space du over dx equals left parenthesis tan to the power of negative 1 end exponent straight x right parenthesis to the power of straight y open square brackets fraction numerator straight y over denominator left parenthesis 1 plus straight x squared right parenthesis tan to the power of negative 1 end exponent straight x end fraction plus dy over dx. log space tan to the power of negative 1 end exponent straight x 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... left parenthesis 2 right parenthesis Also space straight v equals straight y to the power of cot space straight x end exponent space rightwards double arrow space log space straight v equals log space straight y to the power of cot space straight x end exponent space rightwards double arrow space log space straight v equals cot space straight x. log space straight y therefore space 1 over straight v dv over dx equals cot space straight x.1 over straight y dy over dx minus cosec squared straight x. space log space straight y therefore space dv over dx equals straight y to the power of cot space straight x end exponent open square brackets fraction numerator cot space straight x over denominator straight y end fraction dy over dx minus cosec squared straight x. log space straight y 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 space space space space space space space space space space... left parenthesis 3 right parenthesis
From (1), (2) and (3), we get,
left parenthesis tan to the power of negative 1 end exponent straight x right parenthesis to the power of straight y open square brackets fraction numerator straight y over denominator left parenthesis 1 plus straight x squared right parenthesis tan to the power of negative 1 end exponent straight x end fraction plus dy over dx. log space tan to the power of negative 1 end exponent straight x close square brackets plus straight y to the power of cot space straight x end exponent open square brackets fraction numerator cot space straight x over denominator straight y end fraction dy over dx minus cosec squared straight x. log space straight y close square brackets equals 0 open square brackets left parenthesis tan to the power of negative 1 end exponent straight x right parenthesis to the power of straight y. space log space tan to the power of negative 1 end exponent straight x plus straight y to the power of cot space straight x end exponent. fraction numerator cot space straight x over denominator straight y end fraction close square brackets dy over dx equals straight y to the power of cot space straight x end exponent cosec squared straight x space log space straight y minus fraction numerator straight y left parenthesis tan to the power of negative 1 end exponent straight x right parenthesis to the power of straight y over denominator left parenthesis 1 plus straight x squared right parenthesis tan to the power of negative 1 end exponent straight x end fraction space space space space space space space space therefore space dy over dx equals fraction numerator straight y to the power of cot space straight x end exponent cosec squared straight x space log space straight y minus begin display style fraction numerator straight y left parenthesis tan to the power of negative 1 end exponent straight x right parenthesis to the power of straight y over denominator left parenthesis 1 plus straight x squared right parenthesis tan to the power of negative 1 end exponent straight x end fraction end style over denominator left parenthesis tan to the power of negative 1 end exponent straight x right parenthesis to the power of straight y. space log space tan to the power of negative 1 end exponent straight x plus straight y to the power of cot space straight x end exponent. begin display style fraction numerator cot space straight x over denominator straight y end fraction end style end fraction

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