# 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 $\dfrac \d {\d z} \left({c f + w g}\right)$ denote the derivative of $c f + w g$.

Then:

$\dfrac \d {\d z} \left({c f + w g}\right) \left({z}\right) = c \dfrac \d {\d z} f \left({z}\right) + w \dfrac \d {\d z} g \left({z}\right)$

for all $z \in D$.

## Proof

 $\displaystyle c \dfrac \d {\d z} f \left({z}\right) + w \dfrac \d {\d z} g \left({z}\right)$ $=$ $\displaystyle \dfrac \d {\d z} \left({c f}\right) \left({z}\right) + \dfrac \d {\d z} \left({w g}\right) \left({z}\right)$ Combination Theorem for Complex-Differentiable Functions: Multiple Rule $\displaystyle$ $=$ $\displaystyle \dfrac \d {\d z} \left({c f + w g}\right) \left({z}\right)$ Combination Theorem for Complex-Differentiable Functions:Sum Rule

$\blacksquare$