# Combination Theorem for Complex Derivatives

## Theorem

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

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

Let $z \in D$.

Let $w, c \in \C$ be arbitrary complex numbers.

Then the following results hold:

### Sum Rule

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

### Multiple Rule

$\left({w f}\right)' \left({z}\right) = w f' \left({z}\right)$

### Combined Sum Rule

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

Then:

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

### Product Rule

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

### Quotient Rule

For all $z \in D$ with $\map g z \ne 0$:

$\map {\paren {\dfrac f g}'} z = \dfrac {\map {f'} z \map g z - \map f z \map {g'} z} {\paren {\map g z}^2}$

## Also see

These theorems show that identical results hold for derivatives of real functions.