Question

How do you find the 1000th derivative of `y=xe^-x`?

Answer 1

`y^((1000))=-1000e^-x+xe^-x`

Explanation:

Try to determine a general pattern for the derivative of the function by taking the first few derivatives.

`y'=e^-x-xe^-x`
`y''=-e^-x-e^-x+xe^-x=-2e^-x+xe^-x`
`y'''=2e^-x+e^-x-xe^-x=3e^-x-xe^-x`
`y^((4))=-3e^-x-e^-x+xe^-x=-4e^-x+xe^-x`

So, we have a prety good pattern going.

The `nth` derivative, where `n` is even ( `n=2, 4,...`), follows the pattern

`y^((n))=-n e^-x+xe^-x`

We want `y^((1000)),` the above pattern is enough for us, no need to try to determine the odd pattern as well.

`y^((1000))=-1000e^-x+xe^-x`