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Continuity And Differentiability

Question

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Differentiate (x² + 7 x + 2) (e^x - sin x) w.r.t.x.

Solution

Let space space space space space straight y equals left parenthesis straight x squared plus 7 straight x plus 2 right parenthesis left parenthesis straight e to the power of straight x minus sin space straight x right parenthesis therefore space dy over dx equals straight d over dx left square bracket left parenthesis straight x squared plus 7 straight x plus 2 right parenthesis left parenthesis straight e to the power of straight x minus sin space straight x right parenthesis right square bracket space space space space space space space space space space space space space equals left parenthesis straight x squared plus 7 straight x plus 2 right parenthesis straight d over dx left parenthesis straight e to the power of straight x minus sin space straight x right parenthesis plus left parenthesis straight e to the power of straight x minus sin space straight x right parenthesis straight d over dx left parenthesis straight x squared plus 7 straight x plus 2 right parenthesis space space space space space space space space space space space space space equals left parenthesis straight x squared plus 7 straight x plus 2 right parenthesis left parenthesis straight e to the power of straight x minus cos space straight x right parenthesis plus left parenthesis straight e to the power of straight x minus sin space straight x right parenthesis left parenthesis 2 straight x plus 7 right parenthesis space space space space space space space space space space space space space equals straight x squared left parenthesis straight e to the power of straight x minus cos space straight x right parenthesis plus 7 straight x left parenthesis straight e to the power of straight x minus cos space straight x right parenthesis plus 2 left parenthesis straight e to the power of straight x minus cos space straight x right parenthesis plus 2 straight x left parenthesis straight e to the power of straight x minus sin space straight x right parenthesis plus 7 left parenthesis straight e to the power of straight x minus sin space straight x right parenthesis space space space space space space space space space space space space space equals straight x squared left parenthesis straight e to the power of straight x minus cos space straight x right parenthesis plus straight x left parenthesis 9 straight e to the power of straight x minus 7 cos space straight x minus 2 sin space straight x right parenthesis plus 9 straight e to the power of straight x minus 2 cos space straight x minus 7 sin space straight x.

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