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Continuity And Differentiability

Question

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Differentiate the following w.r.t.x: $fraction numerator straight e to the power of straight x over denominator sin space straight x end fraction$

Solution

Let space space space space space straight y equals fraction numerator straight e to the power of straight x over denominator sin space straight x end fraction therefore space dy over dx equals fraction numerator sin space straight x. begin display style straight d over dx end style left parenthesis straight e to the power of straight x right parenthesis minus straight e to the power of straight x. begin display style straight d over dx end style left parenthesis sin space straight x right parenthesis over denominator sin squared straight x end fraction space space space space space space space space space space space space space equals fraction numerator sin space straight x. straight e to the power of straight x minus straight e to the power of straight x space cos space straight x over denominator sin squared straight x end fraction equals straight e to the power of straight x open parentheses fraction numerator sin space straight x minus cos space straight x over denominator sin squared straight x end fraction close parentheses

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