Distributional Derivative on Distributions is Linear Operator

Theorem
The Distributional derivative on distributions is a linear operator.

Proof
Let $\phi, \psi \in \map \DD \R$ be test functions.

Let $\alpha \in \C$ be a complex number.

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

Then:

Thus the distributional derivative is a linear operator.