Product Rule for Complex Derivatives

Theorem
Let $\map f z, \map j z, \map k z$ be single-valued continuous complex functions in a domain $D \subseteq \C$, where $D$ is open.

Let $f$, $j$, and $k$ be complex-differentiable at all points in $D$.

Let $\map f z = \map j z \, \map k z$.

Then:
 * $\forall z \in D: \map {f'} z = \map j z \, \map {k'} z + \map {j'} z \, \map k z$

Proof
Let $z_0 \in D$ be a point in $D$.