How to prove that $f'(x) = e^{-x} g(x)$?

Question

Hello guys can someone help me with that?

I tried all the ways but it always leads me far away from the wanted result.

Given $g(x)= e^{x}-x-1$ and $f(x)=x-1+(x+2)e^{-x}$, prove that $f'(x) = e^{-x} g(x)$.

Answer 1

Your answer is correct. We can factorise it as follows; $$f'(x)=1+e^{-x}(-x-1)=e^{-x}(e^x-x-1)=e^{-x}g(x)$$