Question

How do you find the derivative of `x^2*e^-x`?

Answer 1

`(uv)' = u'v+uv'`

`u = x^2`
`u' = 2x`

`v = e^(-x)`
We know that `(e^(ax))' = (ax)'*e^(ax)`.
`ax = -x`
`(-x)' = -1`
`v' = -e^(-x)`

Therefore :

`(uv)' = 2x ( e^(-x)) + x^2(-e^(-x))`

`(uv)' = ( 2x-x^2) (e^(-x))`

`(uv)' = (2-x) xe^(-x)`.