Let $M\subset \mathbb{R}^k$ be a smooth $m$-manifold, and let $X$ be a smooth vector field on $M$. Let $\phi$ be the flow of $X$, defined via $\phi(t,p) := \gamma(t)$, with $\gamma(0) = p_0$ (where $\gamma(t)$ satisfies the equation $\dot{\gamma}(t) = X(\gamma(t))$.)
Let $Y$ be another vector field on $M$, with flow $\psi$. Set $\phi^s(\cdot) := \phi(s,\cdot)$, and $\psi^t$ analogously. Let $\beta(s,t) := \phi^s\circ\psi^t\circ\phi^{-s}\psi^{-t}(p)$, for some $p\in M$.
My professor now writes $$ \frac{\partial\beta}{\partial s}(0,t) = X(p) - d\psi^t(\psi^{-t}(p))X(\psi^{-t}(p))$$ I don't understand exactly how he is using the chain rule, etc., to get to this equation. I know that $$ \frac{d}{dt}\phi^t(p) = X(\phi^t(p)),\quad \phi^0(p)=p,$$ but I can't figure out the rest.
You professor uses the differential of a function defined by two variables:
$\Psi(s,s')=\phi_s\psi_t\phi_{-s'}\psi_{-t}$. Then compute $\partial_s\Psi(0,s')+\partial_{s'}\Psi(s,0)$ which gives the result.