Relating the spatial derivative of a flow and its generating vector field

Question

Let $M$ be a smooth manifold, and let $X$ be a compactly supported vector field on $M$. Let $\psi:M \to M$ be the $1$-time flow associated with $X$.

Is there a reasonable way to express $d\psi$ in terms of $X$?

I am referring to the "spatial" derivative $d\psi_p:T_pM \to T_{\psi(p)}M$, not to the time derivative of the $t$-flow which is given directly by $X$ ($\frac{d}{dt} \psi_t(p) = X(\psi_{t}(p)$)