Chain Rule Along an Affine Path

Let $n\ge1$ be a natural number, let $\mathbb{R}$ be the real numbers, let $U$ be an open subset of Euclidean space $\mathbb{R}^n$, and let $f:U\to\mathbb{R}$, regarded where a map into a Euclidean space is required as the map into $\mathbb{R}^1$ whose single coordinate function is $f$ itself. Let $x=(x_1,\dots,x_n)$ and $h=(h_1,\dots,h_n)$ be points of $\mathbb{R}^n$, and for $\tau\in\mathbb{R}$ write $x+\tau h$ for the point of $\mathbb{R}^n$ whose $k$th coordinate is $x_k+\tau h_k$ for $k\in\{1,\dots,n\}$.

Let $J\subseteq\mathbb{R}$ be an interval with $x+\tau h\in U$ for every $\tau\in J$, let $F:J\to\mathbb{R}$ be given by

and let $\tau_0\in J$ be an interior point of $J$.

Suppose that $f$ is differentiable at $x+\tau_0h$. By claim 1 of A Derivative Matrix is the Jacobian Matrix, and is Unique the partial derivative $\partial_i f$ of $f$ with respect to the $i$th variable then exists at $x+\tau_0h$ for every $i\in\{1,\dots,n\}$.

Then $F$ is differentiable at $\tau_0$, with