Let $m,n \in \mathbb{N}$, let $U \subset \mathbb{R}^m$ be open, and let $a \in U$. If $f,g: U \to \mathbb{R}^n$ are differentiable at $a$ and $\lambda \in \mathbb{R}$, then the maps $f+g: U \to \mathbb{R}^n$ and $\lambda f: U \to \mathbb{R}^n$ are differentiable at $a$, and
If $u,v: U \to \mathbb{R}$ are differentiable at $a$, then the pointwise product $uv: U \to \mathbb{R}$ is differentiable at $a$, and for every $h \in \mathbb{R}^m$,