In Audin and Damian's "Morse Theory and Floer Homology", In Prop. 5.4.5, there is a statement about the time derivative of a pushforward that I am having trouble understanding. In the last two lines of the proof, the following equality is mentioned. $X_H$ is a Hamiltonian vector field with flow $\psi^t$. $Z$ is a vector in the appropriate tangent space. I don't understand how this equality holds.
$\frac{d}{dt}T_x\psi^t(Z) = T_x\psi^t([X_H, Z])$
In fact this holds for an arbitrary vector field $X$ on the manifold. Denoting by $\psi^t$ its flow, recall that $\psi^{t+s}=\psi^t\circ\psi^s$. Then $$\partial_t\left(\mathrm{d}\psi^t(Z)\right)=\partial_{s=0}\left(\mathrm{d}\psi^{t+s}(Z)\right)=\mathrm{d}\psi^t\partial_{s=0}\left(\mathrm{d}\psi^{s}(Z)\right)=\mathrm{d}\psi^t\left([X,Z]\right)$$ where in the last equality we just used the definition of Lie derivative along a vector field, see for example the definition in terms of flows of the Lie derivative given in the Wikipedia page.
Be careful that, depending on your definition of the Lie derivative, this might actually have the opposite sign.