Question

$Differentiate space sin to the power of negative 1 end exponent straight x space straight w. straight r. straight t. space cos to the power of negative 1 end exponent square root of 1 minus straight x squared end root.$

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Solution

Let space space space space space space space space space straight y equals sin to the power of negative 1 end exponent straight x space comma space straight u equals cos to the power of negative 1 end exponent square root of 1 minus straight x squared end root therefore space space space space space dy over dx equals fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction space space space space space space space space space du over dx equals negative fraction numerator 1 over denominator square root of 1 minus left parenthesis 1 minus straight x squared right parenthesis end root end fraction. straight d over dx left parenthesis square root of 1 minus straight x squared end root right parenthesis equals negative 1 over straight x. fraction numerator negative 2 straight x over denominator 2 square root of 1 minus straight x squared end root end fraction equals fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction Now 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 fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction cross times fraction numerator square root of 1 minus straight x squared end root over denominator 1 end fraction equals 1

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