Let $A$ be the set of all infinitely differentiable functions $f:[0,1] \rightarrow \mathbb{R}$, and let $A_0 \subset A$ be the set of all such functions for which the condition $f(0) = 0$ holds. Define the function $D:A_0 \rightarrow A$, $D(f) = df/dx$.
Use the Mean Value Theorem to show that $D$ is injective.
Use the Fundamental Theorem of Calculus to show that $D$ is surjective.