Let $m\in\mathbb{N}$, let $U\subset\mathbb{R}^m$ be open, let $a\in U$, let $i\in\{1,\dots,m\}$, let $f,g:U\to\mathbb{R}$ be functions, and let $\lambda,\mu\in\mathbb{R}$. Suppose that the partial derivatives $\partial_{x_i}f(a)$ and $\partial_{x_i}g(a)$ exist. Then the partial derivatives $\partial_{x_i}(\lambda f+\mu g)(a)$ and $\partial_{x_i}(fg)(a)$ exist, and

\begin{align*} \partial_{x_i}(\lambda f+\mu g)(a)=\lambda\partial_{x_i}f(a)+\mu\partial_{x_i}g(a). \end{align*}

\begin{align*} \partial_{x_i}(fg)(a)=f(a)\partial_{x_i}g(a)+g(a)\partial_{x_i}f(a). \end{align*}