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Continuity And Differentiability

Question

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Differentiate the following functions w.r.t.x: $straight e to the power of sin space straight x end exponent. sine to the power of straight x$

Solution

Let space space space space space straight y equals straight e to the power of sin space straight x end exponent. sine to the power of straight x therefore space dy over dx equals straight e to the power of sin space straight x end exponent. straight d over dx left square bracket sine to the power of straight x right square bracket plus sin space straight e to the power of straight x. straight d over dx left parenthesis straight e space to the power of sin space straight x end exponent right parenthesis space space space space space space space space space space space space space equals straight e to the power of sin space straight x end exponent. cose to the power of straight x straight d over dx left parenthesis straight e to the power of straight x right parenthesis plus sin space straight e to the power of straight x. straight e to the power of sin space straight x end exponent. straight d over dx left parenthesis sin space straight x right parenthesis space space space space space space space space space space space space space equals straight e to the power of sin space straight x end exponent. cos space straight e to the power of straight x. ex plus sin space straight e to the power of straight x. straight e to the power of sin space straight x end exponent. cos space straight x space space space space space space space space space space space space space equals straight e to the power of sin space straight x end exponent left square bracket straight e to the power of straight x space cos space straight e to the power of straight x plus sin space straight e to the power of straight x. cos space straight x right square bracket

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