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Continuity And Differentiability

Question
CBSEENMA12035056
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Find space dy over dx when space left parenthesis cos space straight x right parenthesis to the power of straight y equals left parenthesis cos space straight y right parenthesis to the power of straight x.

Solution
Here space space space space left parenthesis cos space straight x right parenthesis to the power of straight y equals left parenthesis cos space straight y right parenthesis to the power of straight x therefore space space space log left parenthesis cos space straight x right parenthesis to the power of straight y equals log left parenthesis cos space straight y right parenthesis to the power of straight x therefore space straight y. space log left parenthesis cos space straight x right parenthesis equals straight x. space log left parenthesis cos space straight y right parenthesis Differentiating space both space sides space straight w. straight r. straight t. straight x comma space we space get comma space space straight y. fraction numerator negative sin space straight x over denominator cos space straight x end fraction plus log left parenthesis cos space straight x right parenthesis. dy over dx equals straight x. fraction numerator negative sin space straight y over denominator cos space straight y end fraction dy over dx plus log left parenthesis cos space straight y right parenthesis.1 therefore space minus straight y space tan space straight x plus log left parenthesis cos space straight x right parenthesis. dy over dx equals negative straight x space tan space straight y dy over dx plus log left parenthesis cos space straight y right parenthesis therefore space left square bracket log left parenthesis cos space straight x right parenthesis plus straight x space tan space straight y right square bracket dy over dx equals straight y space tan space straight x plus log left parenthesis cos space straight y right parenthesis therefore space dy over dx equals fraction numerator straight y space tan space straight x plus log left parenthesis cos space straight y right parenthesis over denominator log left parenthesis cos space straight x right parenthesis plus straight x space tan space straight y end fraction

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