Let $M$ be a smooth manifold. I want to show that $$\frac{d}{dt}\varphi^*_t\omega=\varphi^*_t L_X\omega\text{ }(*)$$ where $X$ is a smooth vector field on $M$, $\varphi_t$ is a family of diffeomorphism generated by a vector field $X$, $L_X$ is a Lie derivative along $X$, and $\omega$ is some $n$-form on $M$.
The idea is to show that $(*)$ holds for $0$-form. Then use the fact that the pullback and Lie derivative commutes with an exterior derivative $d$ and since $\Omega(M)$ is generated by $0$-form and $1$-form, we will obtain the desired fact.
I struggle to argue that $$d(\frac{d}{dt}\varphi^*_t\omega)=\frac{d}{dt}d\varphi^*_t\omega.$$ Is it obvious? Do I just need to write it down in local coordinates? Is there any source with more details for this argument?