Question

Differentiate the following w.r.t.x: $log open square brackets straight e to the power of 3 straight x end exponent open parentheses fraction numerator 5 straight x minus 3 over denominator 4 straight x plus 2 end fraction close parentheses to the power of 1 third end exponent close square brackets$

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Solution

Let space straight y equals log open square brackets straight e to the power of 3 straight x end exponent open parentheses fraction numerator 5 straight x minus 3 over denominator 4 straight x plus 2 end fraction close parentheses to the power of 1 third end exponent close square brackets equals log space straight e to the power of 3 straight x end exponent plus log open parentheses fraction numerator 5 straight x minus 3 over denominator 4 straight x plus 2 end fraction close parentheses to the power of 1 third end exponent space space space space space space space space equals 3 straight x space log space straight e plus 1 third log open parentheses fraction numerator 5 straight x minus 3 over denominator 4 straight x plus 2 end fraction close parentheses equals 3 straight x plus 1 third left square bracket log left parenthesis 5 straight x minus 3 right parenthesis minus log left parenthesis 4 straight x plus 2 right parenthesis right square bracket therefore space dy over dx equals 3 plus 1 third open square brackets fraction numerator 5 over denominator 5 straight x minus 3 end fraction minus fraction numerator 4 over denominator 4 straight x plus 2 end fraction close square brackets

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