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Continuity And Differentiability

Question
CBSEENMA12035136
Wired Faculty App

Differentiate space straight w. straight r. straight t. straight x colon space straight y equals sin to the power of negative 1 end exponent open parentheses fraction numerator 1 plus 2 sin space straight x over denominator 2 plus cos space straight x end fraction close parentheses

Solution
space space space space space space space space space space space straight y equals sin to the power of negative 1 end exponent open parentheses fraction numerator 1 plus 2 sin space straight x over denominator 2 plus cos space straight x end fraction close parentheses therefore space dy over dx equals fraction numerator 1 over denominator square root of 1 minus open parentheses begin display style fraction numerator 1 plus 2 sin space straight x over denominator 2 plus cos space straight x end fraction end style close parentheses squared end root end fraction. straight d over dx open parentheses fraction numerator 1 plus 2 sin space straight x over denominator 2 plus cos space straight x end fraction close parentheses space space space space space space space space space space space space space equals fraction numerator 2 plus cos space straight x over denominator square root of left parenthesis 2 plus cos space straight x right parenthesis squared minus left parenthesis 1 plus 2 sin space straight x right parenthesis end root end fraction. fraction numerator left parenthesis 2 plus cos space straight x right parenthesis. left parenthesis 2 cos space straight x right parenthesis minus left parenthesis 1 plus 2 sin space straight x right parenthesis left parenthesis negative sin space straight x right parenthesis over denominator left parenthesis 2 plus cos space straight x right parenthesis squared end fraction space space space space space space space space space space space space space equals fraction numerator 2 plus cos space straight x over denominator square root of 4 plus cos squared straight x plus 4 cos space straight x minus 1 minus 4 sinx minus 4 sin squared straight x end root end fraction cross times fraction numerator 4 cos space straight x plus 2 cos squared straight x plus sin space straight x plus 2 sin squared straight x over denominator left parenthesis 2 plus cos space straight x right parenthesis squared end fraction space space space space space space space space space space space space space equals fraction numerator 1 over denominator square root of 3 plus cos squared straight x minus 4 sin squared straight x plus 4 cos space straight x minus 4 sin space straight x end root end fraction cross times fraction numerator 4 cos space straight x plus sin space straight x plus 2 left parenthesis cos squared straight x plus sin squared straight x right parenthesis over denominator 2 plus cos space straight x end fraction space space space space space space space space space space space space space equals fraction numerator 4 cos space straight x plus sin space straight x plus 2 over denominator left parenthesis 2 plus cos space straight x right parenthesis square root of 3 plus cos squared straight x minus 4 sin squared straight x plus 4 cos space straight x minus 4 sin space straight x end root end fraction

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