Derivatives of the Slice of a Function Along a Line

Let $n\ge1$ be a natural number, let $\mathbb{R}$ be the real numbers, and let $U$ be an open subset of Euclidean space $\mathbb{R}^n$. Write $z\cdot z'$ for the dot product of points of $\mathbb{R}^n$ and $Av$ for the matrix-vector product.

Let $f:U\to\mathbb{R}$ be of class $C^1$ on $U$ (via clause 3 there), and write $\partial_i f$ for its partial derivative with respect to the $i$th variable.

Let $p=(p_1,\dots,p_n)$ and $h=(h_1,\dots,h_n)$ be points of $\mathbb{R}^n$, and for $t\in\mathbb{R}$ write $p+t\,h$ for the point of $\mathbb{R}^n$ whose $k$th coordinate is $p_k+t\,h_k$ for $k\in\{1,\dots,n\}$. Let $J\subseteq\mathbb{R}$ be an interval with $p+t\,h\in U$ for every $t\in J$, and let $g:J\to\mathbb{R}$ be the slice given by

$$g(t)=f(p+t\,h).$$

Let $t_0$ be an interior point of $J$ and write $x_0=p+t_0\,h$. Derivatives of real functions are those of the one-dimensional derivative at an interior point, well defined by Uniqueness of the Derivative at an Interior Point.

Then the following hold.

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

$$g'(t_0)=\sum_{i=1}^{n}\partial_i f(x_0)\,h_i=h\cdot Df(x_0).$$

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

$$g_1(t)=\sum_{i=1}^{n}\partial_i f(p+t\,h)\,h_i ,$$

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 $D^2f(x_0)$,

$$g_1'(t_0)=h\cdot\bigl(D^2f(x_0)\,h\bigr).$$