Question

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

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Solution

Let space space space space space straight y equals fraction numerator sin 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 sin to the power of negative 1 end exponent right parenthesis minus left parenthesis sin to the power of negative 1 end exponent right parenthesis. begin display style straight d over dx end style left parenthesis square root of 1 minus straight x squared end root right parenthesis over denominator left parenthesis square root of 1 minus straight x squared end root right parenthesis 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. begin display style fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction end style minus left parenthesis sin to the power of negative 1 end exponent right parenthesis cross times 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 1 plus begin display style fraction numerator straight x space sin 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 equals fraction numerator square root of 1 minus straight x squared end root plus straight x space sin to the power of negative 1 end exponent straight x over denominator left parenthesis 1 minus straight x squared right parenthesis to the power of bevelled 3 over 2 end exponent end fraction

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Continuity And Differentiability

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

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