Chapter 1.5, Problem 1.46P

(a) Show that $x\frac{d}{dx}\left(\delta \left(x\right)\right)=-\delta \left(x\right)$ .
[Hint: Use integration by parts.]
(b) Let $\theta \left(x\right)$ be the step function: $\theta \left(x\right)=\left\{\begin{array}{cc}1,& \text{if}x>0\\ 0,& \text{if}x\le 0\end{array}\right\}$ .    (1.95)
Show that $d\theta /dx=\delta \left(x\right)$ .