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:
 * $D_x \left({f \left({c x}\right)}\right) = c D_{c x} \left({f \left({c x}\right)}\right)$

Proof
First it is shown that $D_x \left({c x}\right) = c$:

Next: