Combination Theorem for Complex Derivatives/Combined Sum Rule

Theorem
Let $D$ be an open subset of the set of complex numbers.

Let $f, g: D \to \C$ be complex-differentiable functions on $D$.

Let $c, w \in \C$.

Let $\map {\dfrac \d {\d z} } {c f + w g}$ denote the derivative of $c f + w g$.

Then:
 * $\map {\map {\dfrac \d {\d z} } {c f + w g} } z = c \dfrac \d {\d z} \map f z + w \dfrac \d {\d z} \map g z$

for all $z \in D$.