Combination Theorem for Complex Derivatives/Combined Sum 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 $c, w \in \C$.

When $\dfrac{\mathrm d}{\mathrm dz} \left({cf + wg}\right)$ denotes the derivative of $cf + wg$, we have:


 * $\dfrac{\mathrm d}{\mathrm dz} \left({cf + wg}\right) \left({z}\right) = c\dfrac{\mathrm d}{\mathrm dz}f \left({z}\right) + w\dfrac{\mathrm d}{\mathrm dz}g \left({z}\right)$

for all $z \in D$.