Derivative of Function of Constant Multiple

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

Let $$c \in \reals$$ 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 we show $$D_x \left({c x}\right) = c$$:

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Next:

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