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Continuity And Differentiability

Question

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Differentiate the following w.r.t.x: $fraction numerator straight e to the power of ax over denominator sin left parenthesis bx plus straight c right parenthesis end fraction$

Solution

Let space space space space space straight y equals fraction numerator straight e to the power of ax over denominator sin left parenthesis bx plus straight c right parenthesis end fraction therefore space dy over dx equals fraction numerator sin left parenthesis bx plus straight c right parenthesis. begin display style straight d over dx end style left parenthesis straight e to the power of ax right parenthesis minus straight e to the power of ax. begin display style straight d over dx end style left square bracket sin left parenthesis bx plus straight c right parenthesis right square bracket over denominator left square bracket sin left parenthesis bx plus straight c right parenthesis right square bracket squared end fraction space space space space space space space space space space space space space equals fraction numerator sin left parenthesis bx plus straight c right parenthesis. ae to the power of ax minus straight e to the power of ax. cos left parenthesis bx plus straight c right parenthesis. straight b over denominator sin squared left parenthesis bx plus straight c right parenthesis end fraction space space space space space space space space space space space space space equals fraction numerator straight e to the power of ax left square bracket sin left parenthesis bx plus straight c right parenthesis minus bcos left parenthesis bx plus straight c right parenthesis right square bracket over denominator sin squared left parenthesis bx plus straight c right parenthesis end fraction

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