Derivative of a Finite Linear Combination of Real Functions

Let $\mathbb{R}$ be the real numbers, let $I\subseteq\mathbb{R}$ be order-convex, and let $x_0\in I$ satisfy $u<x_0<v$ for some $u,v\in I$, so that $x_0$ is an interior point of $I$ and derivatives at $x_0$ in the sense of Derivative at an Interior Point are defined.

Let $m$ be a natural number, let $[m]$ be the initial segment determined by $m$, and for each $k\in[m]$ let $c_k\in\mathbb{R}$ and let $g_k:I\to\mathbb{R}$ be differentiable at $x_0$, the value $g_k'(x_0)$ being well defined by Uniqueness of the Derivative at an Interior Point. Let $G:I\to\mathbb{R}$ be the function whose value at $x$ is the finite sum

Then $G$ is differentiable at $x_0$ and