Hint
Use the product rule. If we write $f(x,y)=g(x,y)h(x,y)$, with $g(x,y)=x+y$ and $h(x,y)=e^x$, then the product rule is just like for ordinary derivatives. For the derivative with respect to $x$, the product rule is:
\begin{align*}
\pdiff{f}{x} &= \pdiff{g}{x}\cdot h(x,y)+g(x,y)\cdot \pdiff{h}{x}.
\end{align*}
Similarly, the derivative with respect to $y$ is
\begin{align*}
\pdiff{f}{y} &= \pdiff{g}{y}\cdot h(x,y)+g(x,y)\cdot\pdiff{h}{y}
\end{align*}
Also, since $h(x,y)$ does not really depend on $y$, we view it as a constant when differentiating with respect to y. This means that $\pdiff{h}{y}$ is what?
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