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:

$\map {\dfrac \d {\d x} } {c \map f x} = c \map {\dfrac \d {\d x} } {\map f x}$


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:

$\forall z \in D : \map {D_z} {c \map f z} = c \map {D_z} {\map f z}$