Product Rule for Distributional Derivatives of Distributions multiplied by Smooth Functions

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

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

Then in the distributional sense it holds that:


 * $\paren {\alpha T}' = \alpha' T + \alpha T'$

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

By the Product Rule for Derivatives:


 * $\paren {\alpha \phi}' = \alpha' \phi + \alpha \phi'$

Hence: