Let $U \subset \mathbb{R}^m$ be open, let $a \in U$, let $f,g: U \to \mathbb{R}^n$ be maps, and let $\alpha,\beta \in \mathbb{R}$. Assume that there is an open neighbourhood $V \subset U$ of $a$ on which $f$ and $g$ are differentiable, and that the derivative maps $x \mapsto Df_x$ and $x \mapsto Dg_x$, viewed as maps from $V$ to the [vector space](/page/Vector%20Space) of linear maps $\mathbb{R}^m \to \mathbb{R}^n$, are differentiable at $a$. Define the bilinear second derivatives by

\begin{align*} D^2f_a(h,k)=(D(x \mapsto Df_x)_a(h))(k) \end{align*}

and

\begin{align*} D^2g_a(h,k)=(D(x \mapsto Dg_x)_a(h))(k) \end{align*}

for all $h,k \in \mathbb{R}^m$. Then the map $\alpha f+\beta g: U \to \mathbb{R}^n$ is twice differentiable at $a$, and its [second derivative](/page/Second%20Derivative) satisfies

\begin{align*} D^2(\alpha f+\beta g)_a(h,k)=\alpha D^2f_a(h,k)+\beta D^2g_a(h,k) \end{align*}

for all $h,k \in \mathbb{R}^m$.