  Let $R$ be a subring of $\mathbb{C}[[z]]$, and let $X \in \mathbb{C}[[z]]$.
The Newton-Puiseux Theorem implies that if the coefficients of $X$ grow
sufficiently rapidly relative to the coefficients of the series in $R$, then
$X$ is transcendental over $R$. We prove an alternative proof of this result by
establishing a relationship between the coefficients of $A(X)$ and
$A^\prime(X)$, where $A(T)$ is a polynomial over $\mathbb{C}[[z]]$.
