Let $n\in\mathbb N$, let $U \subset \mathbb R^n$ be open, and let $f,g: U \to \mathbb R$ be functions in $C^1(U;\mathbb R)$. Define $f+g:U\to\mathbb R$ by $(f+g)(x)=f(x)+g(x)$, define $fg:U\to\mathbb R$ by $(fg)(x)=f(x)g(x)$, and, when $g(x)\neq 0$ for every $x\in U$, define $f/g:U\to\mathbb R$ by $(f/g)(x)=f(x)/g(x)$. Then $f+g \in C^1(U;\mathbb R)$ and $fg \in C^1(U;\mathbb R)$, with $\nabla(f+g)=\nabla f+\nabla g$ and $\nabla(fg)=f\nabla g+g\nabla f$. If $g(x)\neq 0$ for every $x\in U$, then $f/g \in C^1(U;\mathbb R)$, with $\nabla(f/g)=(g\nabla f-f\nabla g)/g^2$.