Question

Differentiate the following functions w.r.t.x: $sin to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator square root of 1 plus straight x squared end root end fraction close parentheses$

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Solution

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

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