Problem
Problem 2.42 Let $X$ and $Y$ be vector fields on a $C^{\infty}$ manifold $M$. Prove that if $\varphi_{t}$ is the local 1-parameter group generated by $X$, we have for all $p \in M$ :
$$
\varphi_{s *}\left(\left(L_{X} Y\right)_{\varphi_{s}^{-1}(p)}\right)=\lim _{t \rightarrow 0} \frac{1}{t}\left(\varphi_{s *} Y_{\varphi_{s}^{-1}(p)}-\varphi_{s+t *} Y_{\varphi_{s+t}^{-1}(p)}\right)
$$