State the chain rule for the composition of two differentiable functions $f: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ and $g: \mathbb{R}^{n} \rightarrow \mathbb{R}^{p}$.

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be differentiable. For $c \in \mathbb{R}$, let $g(x)=f(x, c-x)$. Compute the derivative of $g$. Show that if $\partial f / \partial x=\partial f / \partial y$ throughout $\mathbb{R}^{2}$, then $f(x, y)=h(x+y)$ for some function $h: \mathbb{R} \rightarrow \mathbb{R}$.