Stimulated by an exercise in da Silva: For a time-dependent vector field $v:M\times \mathbb{R}\to TM$, $k$-form $w:M\to T^{k,0}{M}$ and the isotopy $\rho:M\times \mathbb{R} \to M$ generated by $v$, show that
$\dfrac{d}{dt} \rho^*_t \omega=\rho^*_t \mathcal{L}_{v_t}\omega$
In the middle of my proof, I want to show that both sides of the equation above commute with the exterior derivative $d$. My attempt (for LHS) : because the exterior derivative commutes with pullbacks, $d\rho^*_t=\rho^*_t d$, we know that
$d\dfrac{\rho^*_{t+s}-\rho^*_t}{s}=\dfrac{\rho^*_{t+s}-\rho^*_t}{s}d$
So, if we can commute $\lim_{s\to 0}$ with $d$, we can say the commutativity of LHS.
$\left(\frac{d}{dt}\rho^*_t \right)d=\lim_{s\to 0} d\dfrac{\rho^*_{t+s}-\rho^*_t}{s}\overset{??}{=}\lim_{s\to 0}d\left(\dfrac{\rho^*_{t+s}-\rho^*_t}{s}\right)=d\dfrac{d}{dt}\rho^*_t$
For this to be true, I think the continuity of $d:\Omega^k(M)\to \Omega^{k+1}(M)$ should be verified.
(the topology of $\Omega^{k}(M)$is the subspace topology induced from $C^{\infty}(M,\mathcal{T}^k_0(M))$, $\Omega^{k}(M)$ is the set of $k$-forms on $M$ and $\mathcal{T}^p_q(M)$ is the $(p,q)$-tensor bundle over $M$)
So, query : My reasoning can be justifiable?
Edit : I found out that I can verify the commutativity in this setting; it is equivalent to the commutativity of partial derivatives.
But I think the continuity of $d$ is still an issue in itself.
Is there any reference or paper on this result?