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Continuity And Differentiability

Question

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Differentiate $square root of fraction numerator straight x minus 1 over denominator straight x plus 2 end fraction end root$ W.r.t x.

Solution

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

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