Derivative of Function of Constant Multiple

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

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

Then:
 * $\map {D_x} {\map f {c x} } = c \map {D_{c x} } {\map f {c x} }$

Proof
First it is shown that $\map {D_x} {c x} = c$:

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