Question

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

Answer 1

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

Explanation:

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)`