Let $$\mathcal X:=(X_{s, t})_{s,t\in \mathbb R}$$ be a smooth family of vector fields on some good enough manifold $M$. Heuristically, we think of $t$ as time and $s$ as some extra parameter. Then we consider the evolution diffeomorphism or the flow $$ \phi_s^{t,t_0} $$ of $\mathcal X$ starting from time $t_0$ to time $t$ but $s$ being fixed. It should be known that we have $$\tag{1} \frac{d}{dt}\phi_s^{t,t_0}=X_{s,t}(\phi^{t,t_0}_s(p)) $$ (to see this, we can actually drop $s$ here). Actually, the equation (1) above is just the equation of the integral curve, which is easy to understand.
However, I encounter a significant trouble if I would like to differentiate in $s$, although it seems to be an elementary problem. Explicitly, given any fixed $t_0$ and $t_1$, we have a smooth family of diffeomorphism $\phi_s^{t_1,t_0}$ parametrized by $s$. Then, how do we compute the following $$ \frac{d}{ds}\phi_s^{t_1,t_0}=? $$ Thank you!