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Continuity And Differentiability

Question

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Use Chain rule to find the derivative of $open parentheses fraction numerator 3 space straight x minus 1 over denominator 2 space straight x plus 1 end fraction close parentheses squared$

Solution

Let space straight y equals open parentheses fraction numerator 3 space straight x minus 1 over denominator 2 space straight x plus 1 end fraction close parentheses squared equals straight u squared space where space straight u equals fraction numerator 3 space straight x minus 1 over denominator 2 space straight x plus 1 end fraction therefore space space space dy over du equals 2 space straight u space space space space space space space du over dx equals fraction numerator left parenthesis 2 straight x plus 1 right parenthesis. begin display style straight d over dx end style left parenthesis 3 straight x minus 1 right parenthesis minus left parenthesis 3 straight x minus 1 right parenthesis. begin display style straight d over dx end style left parenthesis 2 straight x plus 1 right parenthesis over denominator left parenthesis 2 straight x plus 1 right parenthesis squared end fraction space space space space space space space space space space space space space space space equals fraction numerator left parenthesis 2 straight x plus 1 right parenthesis.3 minus left parenthesis 3 straight x minus 1 right parenthesis.2 over denominator open parentheses 2 straight x plus 1 close parentheses squared end fraction equals fraction numerator 6 straight x plus 3 minus 6 straight x plus 2 over denominator left parenthesis 2 straight x plus 1 right parenthesis squared end fraction therefore space space du over dx equals fraction numerator 5 over denominator left parenthesis 2 straight x plus 1 right parenthesis squared end fraction By space Chain space Rule comma space space space space space dy over du equals dy over du dy over dx equals left parenthesis 2 space straight u right parenthesis open square brackets fraction numerator 5 over denominator left parenthesis 2 straight x plus 1 right parenthesis squared end fraction close square brackets equals fraction numerator 10 over denominator left parenthesis 2 straight x plus 1 right parenthesis squared end fraction cross times fraction numerator 3 straight x minus 1 over denominator 2 straight x plus 1 end fraction space space space space space space space space space space space space equals fraction numerator 10 left parenthesis 3 straight x minus 1 right parenthesis over denominator left parenthesis 2 straight x plus 1 right parenthesis cubed end fraction

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