Let $\phi:\mathbb R \to \mathbb R$ be a test function. We denote $D(\mathbb R)$ the set of test functions. The dirac distribution $$\delta :D(\mathbb R)\to \mathbb R$$ is defined by: $$<\delta , \phi>=\phi(0)$$ Now take any smooth function $g:\mathbb R\to \mathbb R$, and define the product distribution $g.\delta$ by: $$<g.\delta,\phi>=<\delta,g\phi>$$ From this definition we see that $$<g.\delta,\phi>=g(0)\phi(0)$$
Now i'm reading the following problem prove that $x\delta(x)=0$ and $x\delta'(x)=-\delta(x)$ wihtout any firther information, and I want to understand the meaning of these formulas regarding the definitions above, I mean what is $x$ and what is $\delta(x)$ and what is the product $x\delta(x)$? It seems like he takes $g$ for being the identity $id :\mathbb R\to \mathbb R$ but this would give $$<id.\delta, \phi>=id(0)\phi(0)=0*\phi(0)=0$$ but i don't see where $x\delta(x)=0$ come from. thank you for your help!