Let $U\subset\mathbb{C}$ be an open subset of the complex plane.
Consider $C^\omega(U)$, the space of all functions $f:U\rightarrow \mathbb{C}$ such that $f$ has a convergent power series representation on $U$, that is
$$f(z) = \sum_{n=0}^\infty c_n z^n$$
for all $z\in U$, where $c_n \in\mathbb{C}$.
Define an operator $D:C^\omega(U)\rightarrow C^\omega(U)$ that satisfies the following conditions.
$1.$ $D$ is not the zero operator: there exists $g\in C^\omega(U)$ such that $Dg\neq 0$ (where $0:U\rightarrow\mathbb{C},z\mapsto 0$ for all $z\in U$ is the zero function).
$2.$ $D$ is linear: for any $f,g\in C^\omega(U), \alpha, \beta \in \mathbb{C}$ we have $D(\alpha f + \beta g) = \alpha Df + \beta Dg$.
$3.$ $D$ satisfies the product rule: for any $f,g\in C^\omega(U)$ we have $D(fg) = (Df)g + f(Dg)$.
Does it necessarily follow that $D$ be the derivative operator? If yes, how can this be proved? If no, can one add other properties of the known derivative as to be satisfied (for example, the chain rule) to uniquely fix $D$ as the derivative?
I have found a couple of questions that deal with the uniqueness of the derivative defined as the usual limit, which follows easily enough by proving that a limit, if one exists, is unique. However, I have not been able to find references on whether such an algebraic definition of a derivative is unique and/or can be made such.