Let $E \subset \mathbb{R}^n$ be open and let $V \in C^1(E;\mathbb{R}^n)$. Let $x_0\in E$, and suppose the solution through $x_0$ is defined on an interval $J \subset \mathbb{R}$ containing $0$. If $K\subset J$ is a compact interval containing $0$, then there is an open neighbourhood $U\subset E$ of $x_0$ such that the flow $\varphi(t,y)$ is defined for every $(t,y)\in K\times U$ and the set $\{\varphi(t,y):t\in K,\ y\in U\}$ has compact closure in $E$. For each $t\in K$, the map $y \mapsto \varphi(t,y)$ is differentiable at $x_0$, and its derivative with respect to the initial state is given by the solution of the variational equation along $t \mapsto \varphi(t,x_0)$.