Smooth Real Function times Derivative of Dirac Delta Distribution

Theorem
Let $\alpha \in \map {C^\infty} \R$ be a smooth real function.

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

Then in the distributional sense it holds that:


 * $\alpha \cdot \delta' = \map \alpha 0 \delta' - \map {\alpha'} 0 \delta$

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