Solution
= Solution
Using the definition of a <distributional derivative>,
$$
\langle x\delta',f\rangle=\langle\delta',xf\rangle=-(xf)'\big|_{x=0}=-f(0)=\langle-\delta,f\rangle.
$$
Hence \b[$x\delta'(x)=-\delta(x)$] as distributions.
= Solution
Using the definition of a <distributional derivative>,
$$
\langle x\delta',f\rangle=\langle\delta',xf\rangle=-(xf)'\big|_{x=0}=-f(0)=\langle-\delta,f\rangle.
$$
Hence \b[$x\delta'(x)=-\delta(x)$] as distributions.