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Continuity And Differentiability

Question
CBSEENMA12035273
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Differentiate space sin to the power of negative 1 end exponent open parentheses fraction numerator 1 minus straight x over denominator 1 plus straight x end fraction close parentheses space straight w. straight r. straight t. space square root of straight x

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

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