Holomorphic Functions and Their Derivatives

In the setting of Complex Analysis: Standing Notation, let $U$ be an open subset of $(\mathbb{C},d_{\mathbb{C}})$, let $f:U\to\mathbb{C}$ be a function, and let $W\subseteq U$ be open in $(\mathbb{C},d_{\mathbb{C}})$.

(i) Holomorphic functions. $f$ is holomorphic on $W$ if it is complex differentiable at every point of $W$.

(ii) The derivative. If $f$ is holomorphic on $W$, its derivative on $W$ is the function $f':W\to\mathbb{C}$, $w\mapsto f'(w)$ (Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (v)), with $f'(w)$ the complex derivative of $f$ at $w$ of The Complex Derivative at a Point (ii).