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Continuity And Differentiability

Question
CBSEENMA12035131
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Differentiate : fraction numerator straight x space cos to the power of negative 1 end exponent straight x over denominator square root of 1 minus straight x squared end root end fraction comma space straight x not equal to 1 comma space straight w. straight r. straight t. straight x

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

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