Proof of Derivative of a Finite Linear Combination of Real Functions

We argue by Principle of Induction for the Natural Numbers on the statement $P(m)$: for every family of constants and of functions indexed by $[m]$ as in the hypotheses, the conclusion holds. Claim 1 of Properties of Finite Sums is used in its two forms, restriction and recursion, the latter giving $\sum_{k=1}^{1}a_k=a_1$ and $\sum_{k=1}^{S(j)}a_k=\bigl(\sum_{k=1}^{j}a_k\bigr)+a_{S(j)}$ for the successor map $S$ of Natural Numbers; claim 2 of Sum and Product Rules for One-Dimensional Derivatives and Continuity is used for sums and constant multiples of functions differentiable at $x_0$; and Uniqueness of the Derivative at an Interior Point identifies the resulting derivative values.

Base case $m=1$. By recursion, $G(x)=c_1\,g_1(x)$ for every $x\in I$, so $G$ is the constant multiple $c_1g_1$. By claim 2 of Sum and Product Rules for One-Dimensional Derivatives and Continuity it is differentiable at $x_0$ with derivative $c_1\,g_1'(x_0)$, which by recursion again is $\sum_{k=1}^{1}c_k\,g_k'(x_0)$.

Induction step. Assume $P(m)$, and let constants $c_k$ and functions $g_k$ be given for $k\in[S(m)]$. Let $H:I\to\mathbb{R}$ be the function whose value at $x$ is $\sum_{k=1}^{m}c_k\,g_k(x)$, formed from the restrictions of the two families to $[m]$; by restriction this is unambiguous. By recursion,

so $G$ is the sum of $H$ and the constant multiple $c_{S(m)}g_{S(m)}$. By $P(m)$ the function $H$ is differentiable at $x_0$ with $H'(x_0)=\sum_{k=1}^{m}c_k\,g_k'(x_0)$, and by claim 2 of Sum and Product Rules for One-Dimensional Derivatives and Continuity the function $c_{S(m)}g_{S(m)}$ is differentiable at $x_0$ with derivative $c_{S(m)}\,g_{S(m)}'(x_0)$. The same claim, applied to the sum, shows that $G$ is differentiable at $x_0$ with

the last equality by recursion applied to the family $k\mapsto c_k\,g_k'(x_0)$. This is $P(S(m))$.

By the principle of induction, $P(m)$ holds for every natural number $m$.