Let $M$ be a smooth manifold and let $Z_t$ be a smooth curve in $T_p M$ - tangent space of $M$ in some $p \in M$. If $h_t(x) = H(t,x)$, where $H: \mathbb R \times M \to \mathbb R$ is smooth, then $$ \left.\frac{\mathrm{d} }{\mathrm{d} t}\right |_{t=0} Z_t(h_t) = \left ( \left.\frac{\mathrm{d} }{\mathrm{d} t}\right |_{t=0} Z_t \right )h_0 + Z_0 \left ( \left. \frac{\mathrm{d} }{\mathrm{d} t}\right |_{t=0} h_t \right ). $$
I'm trying to proof the former identity.
There is a hint to consider the smooth function $F: \mathbb R \times \mathbb R \to \mathbb R$ such that $F(t,0) = p$ and $\left. \frac{\partial }{\partial s}\right|_{s=0} F(t,s) = Z_t$, for all $t \in \mathbb R$ and then apply the chain rule.
The existance of such a function can be guaranteed by the existence and uniqueness theorem for o.d.e., however I don't see how can I apply the chain rule to get the desired result.
Help?