Derivative of Constant Multiple

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Theorem

Real

Let $f$ be a real function which is differentiable on $\R$.

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


Then:

$\map {D_x} {c \, \map f x} = c \, \map {D_x} {\map f x}$


Complex

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:

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