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Continuity And Differentiability

Question
CBSEENMA12034709

Differentiate the following w.r.t.x:square root of fraction numerator 1 minus straight x squared over denominator 1 plus straight x squared end fraction end root

Solution
Let space space space space space space straight y equals square root of fraction numerator 1 minus straight x squared over denominator 1 plus straight x squared end fraction end root equals open parentheses fraction numerator 1 minus straight x squared over denominator 1 plus straight x squared end fraction close parentheses to the power of 1 half end exponent
therefore space dy over dx equals 1 half open parentheses fraction numerator 1 minus straight x squared over denominator 1 plus straight x squared end fraction close parentheses to the power of negative 1 half end exponent straight d over dx open parentheses fraction numerator 1 minus straight x squared over denominator 1 plus straight x squared end fraction close parentheses
space space space space space space space space space space space space space equals 1 half open parentheses fraction numerator 1 plus straight x squared over denominator 1 minus straight x squared end fraction close parentheses to the power of 1 half end exponent. open square brackets fraction numerator left parenthesis 1 plus straight x squared right parenthesis. left parenthesis negative 2 straight x right parenthesis minus left parenthesis 1 minus straight x squared right parenthesis.2 straight x over denominator left parenthesis 1 plus straight x squared right parenthesis squared end fraction close square brackets
space space space space space space space space space space space space space equals fraction numerator left parenthesis 1 plus straight x squared right parenthesis to the power of begin display style 1 half end style end exponent over denominator 2 left parenthesis 1 minus straight x squared right parenthesis to the power of 1 half end exponent end fraction. fraction numerator negative 2 straight x minus 2 straight x cubed minus 2 straight x plus 2 straight x cubed over denominator left parenthesis 1 plus straight x squared right parenthesis squared end fraction
space space space space space space space space space space space space space equals fraction numerator left parenthesis 1 plus straight x squared right parenthesis to the power of begin display style 1 half end style end exponent over denominator 2 left parenthesis 1 minus straight x squared right parenthesis to the power of 1 half end exponent end fraction cross times fraction numerator negative 4 straight x over denominator left parenthesis 1 plus straight x squared right parenthesis squared end fraction equals negative fraction numerator 2 straight x over denominator left parenthesis 1 minus straight x squared right parenthesis to the power of bevelled 1 half end exponent left parenthesis 1 plus straight x squared right parenthesis to the power of bevelled 3 over 2 end exponent end fraction

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