if e^x+e^y=e^x+y,prove that dy/dx= -e^y-x

the given equation is 
$$\begin{gathered} e^x+e^y=e^{x+y} \\ {e}^x+e^y=e^x.e^y \\ \frac{e^x}{e^x.e^y}+\frac{e^y}{e^x.e^y}=1 \\ {e}^{-y}+e^{-x}=1 \end{gathered}$$
differentiating wrt x :
$$\begin{gathered} -e^{-y}*\frac{dy}{dx}-e^{-x}=0 \\ {e}^{-y}*\frac{dy}{dx}=-e^{-x} \\ \frac{dy}{dx}=-e^y.e^{-x}=-e^{y-x} \end{gathered}$$
which is the required result.
hope this helps you