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Continuity And Differentiability

Question

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Differentiate log [(x + 2)(x³ - x)] w.r.t.x

Solution

Let space straight y equals log left square bracket left parenthesis straight x plus 2 right parenthesis left parenthesis straight x cubed minus straight x right parenthesis right square bracket therefore space space straight y equals log left parenthesis straight x plus 2 right parenthesis plus log left parenthesis straight x cubed minus straight x right parenthesis therefore space dy over dx equals fraction numerator 1 over denominator straight x plus 2 end fraction. straight d over dx left parenthesis straight x plus 2 right parenthesis plus fraction numerator 1 over denominator straight x cubed minus straight x end fraction. straight d over dx left parenthesis straight x cubed minus straight x right parenthesis space space space space space space space space space space space space space equals fraction numerator 1 over denominator straight x plus 2 end fraction cross times 1 plus fraction numerator 1 over denominator straight x cubed minus straight x end fraction cross times left parenthesis 3 straight x squared minus 1 right parenthesis equals fraction numerator 1 over denominator straight x plus 2 end fraction plus fraction numerator 3 straight x squared minus 1 over denominator straight x cubed minus straight x end fraction

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