I'm trying to prove the following identity:
$$ \begin{equation} \label{eq:1} x\frac{d}{dx}\delta(x)=-\delta(x) \end{equation} $$
I integrated both sides with respect to $x$ over the limits of $(-\infty,\infty)$ and saw that they both evaluated to $-1$.
I'm not sure if that is the right approach because the worked solution I saw multiplied both sides of the above relation by an arbitrary function $f(x)$.
Is there any reason for multiplying both sides by an arbitrary $f(x)$? I'm a little helpless here because the approach that I took seemed more direct.
By definitions, for any $\varphi \in C_c^\infty(\mathbb R),$ $$ \langle x \delta', \varphi \rangle = \langle \delta', x\varphi \rangle = - \langle \delta, (x\varphi)' \rangle = - \langle \delta, \varphi + x\varphi' \rangle = - (\varphi + x\varphi')|_{x=0} = -\varphi(0) = \langle -\delta, \varphi \rangle . $$ Thus, $x \delta' = -\delta.$