I have to prove the following.
Let $M$ be a differentiable smooth manifold and let $\chi \in \Gamma(TM)$ a smooth vector field on $M$. Denote by $\mathcal{L}_{\chi}$ the Lie derivative along $\chi$ and let $\Theta_t \in C^{\infty}(M \times \mathbb{R}; M \times \mathbb{R}), (p,s) \mapsto \Theta_t(p,s) = (p, s+t)$ be the flow map associated with $\chi$. Prove that $\iota^*_t \mathcal{L}_{\chi} = \frac{d}{dt} \iota^*_t$, where $\iota_t : M \rightarrow M \times \mathbb{R}, p \mapsto (p,t)$.
($\textbf{Hint}$: Note that $\iota_t = \Theta_t \iota_0$).
I am not very confident with the notion of Lie derivative. My idea was to start from the definition of Lie derivative along $\chi$, that is $\mathcal{L}_{\chi} \omega = \frac{d}{dt} \Theta^*_t \omega$ where $\omega \in \Omega^{k}(M \times \mathbb{R})$. However, I don't know how to proceed.
If anyone could give me a suggestion on how to prove this, it would be greatly appreciated.