Let $U \subset \mathbb{R}^m$ be open, let $a \in U$, let $u,v \in \mathbb{R}^m$, let $f,g: U \to \mathbb{R}^n$ be functions, and let $\alpha,\beta \in \mathbb{R}$. Suppose that the iterated directional derivatives $D_uD_v f(a)$ and $D_uD_v g(a)$ exist. Then the [iterated directional derivative](/page/Iterated%20Directional%20Derivative) $D_uD_v(\alpha f+\beta g)(a)$ exists, and

\begin{align*} D_uD_v(\alpha f+\beta g)(a)=\alpha D_uD_v f(a)+\beta D_uD_v g(a). \end{align*}