Dirac Delta identity proof.

Question

I was working on showing that: $$x \frac{\mathrm{d}(\delta (x))}{\mathrm{d}x} = -\delta(x)$$ using integration by parts. I arrived to a point where I had had as an answer the following: $$ -f(0) - \int_{-\infty}^\infty \delta(x)x\frac{\mathrm{d}f}{\mathrm{d}x} \ \mathrm{d}x$$ I would greatly appreciate if someone could please provide an explanation as to why the second term is zero. Thank you in advance.

Answer 1

It is zero because the integrand $$\delta(x) x\frac{df}{dx}$$ is zero for all $x$.

If $x=0$ then it is obviously zero, and if $x\neq 0$ then it is also zero because $\delta(x)=0$.