Product Rule for Complex Derivatives

Theorem
Let $f \left({z}\right), j \left({z}\right), k \left({z}\right)$ 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 $f \left({z}\right) = j \left({z}\right) k \left({z}\right)$.

Then:
 * $\forall z \in D: f' \left({z}\right) = j \left({z}\right) \, k' \left({z}\right) + j' \left({z}\right) \, k \left({z}\right)$

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