Intermediate Value Theorem for Derivatives

Theorem
Let $I$ be an open interval.

Let $f : I \to \R$ be everywhere differentiable.

Then $f'$ satisfies the Intermediate Value Property.

Proof
Since $\forall \set {a, b \in I: a < b}: \openint a b \subseteq I$, the result follows from Image of Interval by Derivative.