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:

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

$$ $$ $$

Next:

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