Question

If y=sin ^-1x, prove that

$dy over dx equals straight x over open parentheses 1 minus straight x squared close parentheses to the power of begin display style 3 over 2 end style end exponent$

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Solution

Here space space straight y equals sin to the power of negative 1 end exponent space straight x therefore space space dy over dx equals fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction equals left parenthesis 1 minus straight x squared right parenthesis to the power of negative 1 half end exponent therefore space space fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative 1 half left parenthesis 1 minus straight x squared right parenthesis to the power of negative 3 over 2 end exponent left parenthesis negative 2 straight x right parenthesis equals fraction numerator straight x over denominator left parenthesis 1 minus straight x squared right parenthesis to the power of 3 over 2 end exponent end fraction

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