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Continuity And Differentiability

Question

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Find f'(x) where f(x)= $square root of fraction numerator 1 minus straight x over denominator 2 plus straight x end fraction end root$

Solution

straight f left parenthesis straight x right parenthesis equals square root of fraction numerator 1 minus straight x over denominator 2 plus straight x end fraction end root equals open parentheses fraction numerator 1 minus straight x over denominator 2 plus straight x end fraction close parentheses to the power of 1 half end exponent Differentiating space straight w. straight r. straight t space straight x comma space we space get straight f apostrophe left parenthesis straight x right parenthesis equals 1 half open parentheses fraction numerator 1 minus straight x over denominator 2 plus straight x 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 over denominator 2 plus straight x end fraction close parentheses space space space space space space space space equals 1 half open parentheses fraction numerator 2 plus straight x over denominator 1 minus straight x end fraction close parentheses to the power of 1 half end exponent open square brackets fraction numerator left parenthesis 2 plus straight x right parenthesis begin display style straight d over dx end style left parenthesis 1 minus straight x right parenthesis minus left parenthesis 1 minus straight x right parenthesis begin display style straight d over dx end style left parenthesis 2 plus straight x right parenthesis over denominator left parenthesis 2 plus straight x right parenthesis squared end fraction close square brackets space space space space space space space space equals 1 half fraction numerator left parenthesis 2 plus straight x right parenthesis to the power of bevelled 1 half end exponent over denominator left parenthesis 1 minus straight x right parenthesis to the power of bevelled 1 half end exponent end fraction open square brackets fraction numerator left parenthesis 2 plus straight x right parenthesis left parenthesis negative 1 right parenthesis minus left parenthesis 1 minus straight x right parenthesis left parenthesis 1 right parenthesis over denominator left parenthesis 2 plus straight x right parenthesis squared end fraction close square brackets equals 1 half fraction numerator left parenthesis 2 plus straight x right parenthesis to the power of bevelled 1 half end exponent over denominator left parenthesis 1 minus straight x right parenthesis to the power of bevelled 1 half end exponent end fraction open square brackets fraction numerator negative 2 minus straight x minus 1 plus straight x over denominator left parenthesis 2 plus straight x right parenthesis squared end fraction close square brackets space space space space space space space space equals 1 half fraction numerator left parenthesis 2 plus straight x right parenthesis to the power of begin display style 1 half end style end exponent over denominator left parenthesis 1 minus straight x right parenthesis to the power of begin display style 1 half end style end exponent end fraction cross times fraction numerator negative 3 over denominator left parenthesis 2 plus straight x right parenthesis squared end fraction equals negative fraction numerator 3 over denominator 2 left parenthesis 1 minus straight x right parenthesis to the power of begin display style 1 half end style end exponent left parenthesis 2 plus straight x right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction

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