(c)

44–54 are about optional topics at the end of the section.

The derivative of $\delta (x)$ is extremely singular. It is a “dipole” known by its integrals. Integrate by parts in (b) and (c):

(a) ${\int }_{-1}^1\frac{d\delta }{dx}dx$

(b) ${\int }_{-1}^{1}x\frac{d\delta }{dx}dx$

(c) ${\int }_{-1}^{1}v(x)\frac{d\delta }{dx}dx=-v^{\prime }(0)$.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §7.1 Integration by Parts, #49