Chain Rule Along an Affine Path

Let $n$ be a natural number, let $U\subseteq\mathbb{R}^{n}$ be an open subset of Euclidean space $\mathbb{R}^{n}$, and let $f:U\to\mathbb{R}$. Let $x,h\in\mathbb{R}^{n}$ with $h=(h_{1},\dots,h_{n})$, addition and scalar multiplication of points of $\mathbb{R}^{n}$ being the coordinatewise operations of Euclidean Space $\mathbb{R}^n$.

Let $p,q\in\mathbb{R}$ be such that $x+\tau h\in U$ for every $\tau$ in the open interval $(p,q)$, and let $F:(p,q)\to\mathbb{R}$ be given by $F(\tau)=f(x+\tau h)$.

Let $\tau_{0}\in(p,q)$ and suppose $f$ is differentiable at $x+\tau_{0}h$. Then $F$ is differentiable at $\tau_{0}$, with

the partial derivatives existing because $f$ is differentiable at that point.