# Lie derivative of a vector field

I'm trying to follow a proof that the Lie derivative of a vector field $Y$ in the direction of a vector field $X$ is the commutator $[X,Y]$. We write $\Phi_t$ for the flow of $X$.

How do I get from $$\frac{d}{dt}\vert _{t=0}\Phi^*_t(Y(\Phi^*_{-t}(f)))$$ to

$$\frac{d}{dt}\vert _{t=0} (\Phi^*_t(Y(f))-Y(\Phi^*_t(f))$$ ? I'm sure it's simple but I'm just not seeing it.

-

Formally, this is because of the product rule (e.g. $\frac{d}{dt}\vert_{t=0} (A(t)BC(t)) = A'(0) B C(0) + A(0) B C'(0)$): $$\frac{d}{dt}\vert_{t=0} \Phi_t^*(Y(\Phi_{-t}^* (f))) = (\frac{d}{dt}\vert_{t=0} \Phi_t^*) (Y(\Phi_0^* f)) + \Phi_0^* Y(\frac{d}{dt}\vert_{t=0} \Phi_{-t}^* f).$$ Now, use the fact that since $\Phi_t$ is a flow, $\Phi_0$ is the identity map and for the second term make the substitution $t \mapsto -t$. The $\frac{d}{dt}\vert_{t=0}$ comes out in the second term since $Y$ is a linear operator and independent of $t$.