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Continuity And Differentiability

Question

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Differentiate the following functions w.r.t.x: $log space sin left parenthesis straight e to the power of straight x plus 5 straight x plus 8 right parenthesis$

Solution

Let space space space space space space straight y equals log space sin left parenthesis straight e to the power of straight x plus 5 straight x plus 8 right parenthesis therefore space space dy over dx equals fraction numerator 1 over denominator sin left parenthesis straight e to the power of straight x plus 5 straight x plus 8 right parenthesis end fraction. straight d over dx left square bracket sin space left parenthesis straight e to the power of straight x plus 5 straight x plus 8 right parenthesis right square bracket space space space space space space space space space space space space space space equals fraction numerator 1 over denominator space sin left parenthesis straight e to the power of straight x plus 5 straight x plus 8 right parenthesis end fraction. cos left parenthesis straight e to the power of straight x plus 5 straight x plus 8 right parenthesis. straight d over dx left parenthesis straight e to the power of straight x plus 5 straight x plus 8 right parenthesis space space space space space space space space space space space space space space equals fraction numerator 1 over denominator space sin left parenthesis straight e to the power of straight x plus 5 straight x plus 8 right parenthesis end fraction. cos left parenthesis straight e to the power of straight x plus 5 straight x plus 8 right parenthesis. left parenthesis straight e to the power of straight x plus 5 right parenthesis space space space space space space space space space space space space space space equals left parenthesis straight e to the power of straight x plus 5 right parenthesis cot left parenthesis straight e to the power of straight x plus 5 straight x plus 8 right parenthesis

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