Derivative of a Sum and of a Difference

Let $\mathbb{R}$ be the set of real numbers, let $p,q\in\mathbb{R}$, let $g,h:(p,q)\to\mathbb{R}$ be functions on the open interval $(p,q)$, and let $c\in(p,q)$. Let $g+h$ and $g-h$ be the functions from $(p,q)$ to $\mathbb{R}$ whose values at $s\in(p,q)$ are $g(s)+h(s)$ and $g(s)+(-h(s))$ respectively.

Suppose $g$ and $h$ are differentiable at $c$. Then $g+h$ and $g-h$ are differentiable at $c$, with