Delta Distributions

Question

I need some help solving the following:

Let $\delta'_0 \in D'\mathbb(R)$ be the distribution given by: $$\delta'_0(u) = -u'(0) \\ u \in D\mathbb(R)$$ Compute $x\delta'_0$.

Answer 1

For any $\phi \in C_C^\infty$, we have

$$\begin{align} \langle x\delta_0',\phi\rangle&=\langle \delta_0',x\phi\rangle\\\\ &=-\langle \delta_0,(x\phi)'\rangle\\\\ &=-\left.\left(x\phi\right)'\right|_{x=0}\\\\ &=-\phi(0) \end{align}$$

Therefore, $x\delta'_0=-\delta_0$.