I am trying to prove the chain rule for formal power series $\mathbb Q[[T]]$, i.e.:
Given $f(T) = \sum_{n \geq 0}f_nT^n, \; h(T) = \sum_{n\geq0}h_nT^n$, then:
$\frac{d}{dT} f(h(T)) = \left(\frac{d}{dT}h(T)\right)\left(\frac{d}{dh(T)}f(h(T))\right)$
Where $\frac{d}{dT}f(T) = \sum_{n\geq 0}nf_{n}T^{n-1}$
I will skip most of the working I have done as it is very long and I believe simple. However I will present the step I am currently stuck on.
I have shown:
$$\frac{d}{dT}f(h(T)) = \sum_{n \geq 0}nT^{n-1}\left( \sum_{m \geq0}f_m\left(\sum_{p_1 + \dots + p_m = n \\ p_i \geq 0}h_{p_1}\dots h_{p_m}\right)\right)$$
And:
$$\left( \frac{d}{dT}h(T)\right)\left(\frac{d}{dh(T)}f(h(T))\right) = \sum_{n \geq 0}T^n\left( \sum_{l \geq 0}f_l\left(l\sum_{p+q = n \\ p,q \geq 0}(p+1)h_{p+1}\left(\sum_{q_1 + \dots + q_{l-1} \\ q_i \geq 0}h_{q_1}\dots h_{q_{l-1}}\right)\right)\right)$$
However, because of the issue of the possibility that the "$h_{p+1}$" might be equal to one of the $h_{q_i}$, it is not straightforward to show that these two expressions are in fact equal.
I wanted to ask if it is indeed possible to show these expressions to be equal, or if perhaps I have made a mistake somewhere in my derivation of these expressions. Any help would be appreciated, thank you.