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

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

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.