Fast-growing series are transcendental

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 $A^{\mathrm{\prime }}(X)$, where $A(T)$ is a polynomial over $\mathbb{C}[[z]]$.