The partial derivative with respect to what variable?

(del F)/(del x) = e^{-x} * (1 - e^{-y}) <--- This is the partial derivative of F(x,y) wrt x.

(del F)/(del y) = (1 - e^{-x}) * e^{-y} <--- This is the partial derivative of F(x,y) wrt y.

neither of which is e^-(x+y). I'm not sure what you are asking for.

-Dan