Smooth Real Function times Derivative of Dirac Delta Distribution/Corollary

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

Then in the distributional sense it holds that:


 * $x \delta' = - \delta$

Proof
From Smooth Real Function times Derivative of Dirac Delta Distribution:


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

where $\alpha$ is a smooth function.

If $\map \alpha x = x$, then:


 * $x \delta' = - \delta$