Linearity of the derivative

Statement

Let $V$ and $W$ be normed vector spaces over $\mathbb{R}$, let $U \subseteq V$ be open, let $f, g \colon U \to W$, let $x \in U$, and let $c \in \mathbb{R}$. Suppose that $f$ and $g$ are Fréchet differentiable at $x$, with Fréchet derivatives $Df(x)$ and $Dg(x)$. Then $f + g$ and $c f$ are Fréchet differentiable at $x$, and $$D(f + g)(x) = Df(x) + Dg(x), \qquad D(c f)(x) = c \, Df(x).$$