Distributional Derivative of Heaviside Step Function

Theorem
Let $H : \R \to \closedint 0 1$ be the Heaviside step function.

Let $T \in \map {\DD'} \R$ be a distribution corresponding to $H$.

Let $\delta \in \map {\DD'} \R$ be the Dirac delta distribution.

Then the distributional derivative of $T$ is $\delta$.

Proof
Let $\phi \in \map \DD \R$ be a test function.

Then: