Distributional Derivative of Heaviside Step Function

Theorem
Let $H \in \map {L^1_\text{loc}} \R$ 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: