Differentiable Function as Distribution

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

Let $f : \R \to \R$ be a continuously differentiable real function.

Suppose $T_f$ is a distribution identified with $f$.

Then $T_f' = T_{f'}$.

Proof
Let $\phi \in \map \DD \R$ be a test function with a support on $\closedint a b$.

Then: