How do you find the derivative for #y= e^-x sinx#?

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Answer 1 · 2018-03-15T16:56:51

#y'(x) = e^(-x)(cos x - sin x)#

Given: #y = e^(-x) sinx#

Use the product rule: #(u*v)' = u* v' + v * u'#

Use the chain rule: #(e^u)' = u' e^u#

Use #(sin x)' = cos x#

Let #u = e^(-x) ;" " v = sin x#

#u' = -e^(-x); " " v' = cos x#

Using the product rule:

#y ' = e^(-x) cos x + sin x (-e^(-x))#

Rearrange: #" "y' = e^(-x) cos x -e^(-x) sin x#

Factor: #" "y' = e^(-x)(cos x - sin x)#