Baby Rudin proves in Theorem 8.1 the differentiability of real power series in their interval of convergence by leveraging his Theorem 7.17:
7.17 Theorem
Suppose $\{f_n\}$ sequence of real-valued functions, differentiable on $[a, b]$ and such that $\{f_n(x_0)\}$ converges for some $x_0$ in $[a, b]$. If $\{f_n'\}$ converges uniformly on $[a, b]$, then $\{f_n\}$ converges uniformly to a function $f$ and $$f' = \lim_{n \to \infty}\{f_n'\}$$ on $[a, b]$.
Examining his proof of Theorem 7.17, it seems that the real-valued Mean Value Theorem is used only to get a bound on differences, and so it seems that Theorem 7.17 should generalize to complex differentiable functions on convex open sets in $\mathbb{C}$.
Motivation:
The proofs that I've seen (Papa Rudin, Ahlfors, Spivak calculus book, and Bak/Newman) that complex power series are termwise differentiable seem to be based one way or another rewriting terms $\frac{z^n - w^n}{z-w}$ as sums and then passing to a limit. I was just wondering if there is another path that avoids the direct calculation.
Question:
Am I correctly understanding that this provides a path to showing that complex power series are termwise differentiable inside their disk of convergence?
Thanks
