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Continuity And Differentiability

Question

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Differentiate $log open parentheses fraction numerator straight x squared plus straight x plus 1 over denominator straight x squared minus straight x plus 1 end fraction close parentheses$

Solution

Let space straight y equals log open parentheses fraction numerator straight x squared plus straight x plus 1 over denominator straight x squared minus straight x plus 1 end fraction close parentheses therefore space straight y equals log left parenthesis straight x squared plus straight x plus 1 right parenthesis minus log left parenthesis straight x squared minus straight x plus 1 right parenthesis dy over dx equals fraction numerator 1 over denominator straight x squared plus straight x plus 1 end fraction cross times straight d over dx left parenthesis straight x squared plus straight x plus 1 right parenthesis minus fraction numerator 1 over denominator straight x squared minus straight x plus 1 end fraction cross times straight d over dx left parenthesis straight x squared minus straight x plus 1 right parenthesis space space space space space space space space equals fraction numerator 1 over denominator straight x squared plus straight x plus 1 end fraction cross times left parenthesis 2 straight x plus 1 right parenthesis equals negative fraction numerator 1 over denominator straight x squared minus straight x plus 1 end fraction cross times left parenthesis 2 straight x minus 1 right parenthesis space space space space space space space space equals fraction numerator 2 straight x plus 1 over denominator straight x squared plus straight x plus 1 end fraction minus fraction numerator 2 straight x minus 1 over denominator straight x squared minus straight x plus 1 end fraction

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