Combination Theorem for Complex Derivatives/Product Rule

Theorem
Let $f, g: D \to \C$ be complex-differentiable functions, where $D$ is an open subset of the set of complex numbers.

Let $f g$ denote the product of the functions $f$ and $g$.

Then $f g$ is complex-differentiable in $D$, and its derivative $\left({f g}\right)'$ is defined by:
 * $\left({f g}\right)' \left({z}\right) = f' \left({z}\right) g \left({z}\right) + f \left({z}\right) g' \left({z}\right)$

for all $z \in D$.

Proof
Denote the open ball of $0$ with radius $r \in \R_{>0}$ as $B_r \left({0}\right)$.

Let $z \in D$.

By the Alternative Differentiability Condition, it follows that there exists $r \in \R_{>0}$ such that for all $h \in B_r \left({0}\right) \setminus \left\{ {0}\right\}$:


 * $f\left({z + h}\right) = f \left({z}\right) + h \left({f' \left({z}\right) + \epsilon_f \left({h}\right) }\right)$
 * $g\left({z + h}\right) = g \left({z}\right) + h \left({g' \left({z}\right) + \epsilon_g \left({h}\right) }\right)$

where $\epsilon_f, \epsilon_g: B_r \left({0}\right) \setminus \left\{ {0}\right\} \to \C$ are continuous functions that converge to $0$ as $h$ tends to $0$.

Then:

Define $\epsilon: B_r \left({0}\right) \setminus \left\{ {0}\right\} \to \C $ by $\epsilon \left({h}\right) = h \left({f' \left({z}\right) + \epsilon_f \left({h}\right) }\right) \left({g' \left({z}\right) + \epsilon_g \left({h}\right) }\right)$.

From Combination Theorem for Continuous Functions/Product Rule and Combined Sum Rule, it follows that $\epsilon$ is continuous.

From Combination Theorem for Limits of Functions/Product Rule and Combined Sum Rule, it follows that:


 * $\displaystyle \lim_{h \to 0} \epsilon \left({h}\right) = \left({\lim_{h \to 0} h}\right) \left({\lim_{h \to 0} \left({f' \left({z}\right) + \epsilon_f \left({h}\right) }\right) }\right) \left({\lim_{h \to 0} \left({g' \left({z}\right) + \epsilon_g \left({h}\right) }\right) }\right)= 0$

Then the Alternative Differentiability Condition shows that:


 * $\left({fg}\right)' \left({z}\right) = f' \left({z}\right) g \left({z}\right) + f \left({z}\right) g' \left({z}\right)$