Derivative of a Coordinate Slice of a $C^1$ Function on a Euclidean Open Set

Let $n\in\mathbb{N}$, let $U\subseteq \mathbb{R}^n$ be open, let $f:U\to\mathbb{R}$ be a $C^1$ map, let $i\in\{1,\dots,n\}$, and let $x=(x_1,\dots,x_n)\in U$. Let $a,b\in\mathbb{R}$ with $a<b$, and assume that

$$\{(x_1,\dots,x_{i-1},t,x_{i+1},\dots,x_n): t\in[a,b]\}\subseteq U.$$

Define $g:[a,b]\to\mathbb{R}$ by

$$g(t)=f(x_1,\dots,x_{i-1},t,x_{i+1},\dots,x_n).$$

Then $g$ is continuous on $[a,b]$, differentiable at every $t\in(a,b)$, and

$$g'(t)=\frac{\partial f}{\partial x_i}(x_1,\dots,x_{i-1},t,x_{i+1},\dots,x_n)$$

for every $t\in(a,b)$.