I have,
e^-(x+y)
I need to find what the derivative is with respect to Y. My answer I found was simply the same as above and I didn't seem to think it was correct.
$\displaystyle e^{-(x+y)}$ can be rewritten as $\displaystyle e^{-x-y}=e^{-x}e^{-y}$ so if you treat e^{-x} as a constant then you just have to differentiate $\displaystyle \frac{\partial}{\partial y} e^{-x}e^{-y} =-e^{-x}e^{-y}
=-e^{-x-y}=-e^{-(x+y)}$
so your final solution should be:
$\displaystyle -e^{-(x+y)}$