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

Corollary
Let $a, b \in \R$ be a constant.

Then:
 * $D_x \left({f \left({a x + b}\right)}\right) = a D_{a x + b} \left({f \left({a x + b}\right)}\right)$.

Proof
First we show $D_x \left({c x}\right) = c$:

Next:

Proof of Corollary
First we show $D_x \left({a x + b}\right) = a$:

Next: