# Derivative of Constant Multiple/Complex

## Theorem

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

Let $f: D \to \C$ be a complex-differentiable function on $D$.

Let $c \in \C$ be a constant.

Then:

$\forall z \in D : D_z \left({c f \left({z}\right)}\right) = c D_z \left({f \left({z}\right)}\right)$

## Proof

 $\displaystyle D_z \left({c f \left({z}\right)}\right)$ $=$ $\displaystyle c D_z \left({f \left({z}\right)}\right) + f \left({z}\right) D_z \left({c}\right)$ Product Rule for Complex Derivatives $\displaystyle$ $=$ $\displaystyle c D_z \left({f \left({z}\right)}\right) + 0$ Complex Derivative of Constant $\displaystyle$ $=$ $\displaystyle c D_z \left({f \left({z}\right)}\right)$

$\blacksquare$