Question

How do you differentiate `f(x)=xe^x-x` using the product rule?

Answer 1

`f'(x)=e^x+xe^x-1`

Explanation:

Let's first examine just the `xe^x` portion of this function.

Through the product rule,
`d/dx[xe^x]=color(blue)(d/dx[x])*e^x+color(purple)(d/dx[e^x]*)x`

Therefore,
`d/dx[xe^x]=color(blue)(1)*e^x+color(purple)(e^x)*x=e^x+xe^x`

We can use this in the original equation to determine that
`f'(x)=e^x+xe^x-d/dx[x]=color(green)(e^x+xe^x-1`