Derivative of Identity Function

Theorem
Let $I_\R: \R \to \R$ be the identity function.

Then $\forall x \in \R: I_\R^{\prime} \left({x}\right) = 1$.

Note that this can be more compactly written $D_x \left({x}\right) = 1$.

Corollary

 * $\dfrac{\mathrm{d}}{\mathrm{d}{x}} \left({c x}\right) = c$

Proof
The identity function is defined as $\forall x \in \R: I_\R \left({x}\right) = x$.

Thus:

Proof of Corollary
Follows directly from the above, and Derivative of Constant Multiple.