Derivatives of the Slice of a Function Along a Line

Let $n\ge1$ be a natural number and let $\mathbb{R}$ be the real numbers. Let $U\subseteq\mathbb{R}^n$ be an open subset of Euclidean space $\mathbb{R}^n$, a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space with the sum of points and the scalar multiple; write $z\cdot z'$ for the dot product.

Let $f:U\to\mathbb{R}$ be of class $C^1$ on $U$, let $p,h\in\mathbb{R}^n$, let $J\subseteq\mathbb{R}$ be order-convex with $p+t\,h\in U$ for every $t\in J$, and let $g:J\to\mathbb{R}$ be the slice given by

Let $t_0\in J$ satisfy $u<t_0<v$ for some $u,v\in J$, so that $t_0$ is an interior point of $J$, and write $x_0=p+t_0\,h$.

Then the following hold, derivatives of real functions being those of Derivative at an Interior Point, well defined by Uniqueness of the Derivative at an Interior Point.

1. (First derivative) $g$ is differentiable at $t_0$ and, with the gradient $Df(x_0)$ and the partial derivatives of $f$,

2. (Second derivative) Suppose in addition that $f$ is of class $C^2$ on $U$, and let $g_1:J\to\mathbb{R}$ be given by

so that by claim 1 the value $g_1(t)$ is $g'(t)$ at every interior point $t$ of $J$. Then $g_1$ is differentiable at $t_0$ and, with the Hessian matrix and the matrix-vector product,