What is the Delta Function Identity?

[ArcherVillage]

I know I haven't entered the formulae with the proper syntax, but I'm extremely exhausted at the time of posting, so please just read it and give advice, forgiving me this once for not using proper form (it's basically in latex code format).

Homework Statement

Show [itex]f(x)\frac{d}{dx}\delta(x) = f(0)\frac{d}{dx}\delta(x)-f'(0)\delta(x)[/itex]

Homework Equations

[tex]\delta(x)\intf(x)dx = \int f(x)\delta(x)dx = f(0)[/tex]
[tex](f',\phi):=(f,-\phi ')[/tex]

The Attempt at a Solution

[tex]f(0)\delta'(x) - f'(0)\delta(x)[/tex]
[tex]=\delta'(x)\intf(x)\delta(x)dx-\delta(x)\intf'(x)\delta(x)dx[/tex]
[tex]=...[/tex]

 

[HallsofIvy]

I would use "integration by parts". To integrate
[tex]\int (f(x) d(\delta(x))/dx) dx[/tex]
let u= f(x) and [itex]dv= d(\delta(x))/dx dx[/itex] so that du= f'(x) dx and [itex]v= \delta(x)[/itex].=

Then [tex]\int (f(x) d(\delta(x))/dx) dx= f'(x)\delta(x)- \int f'(x)\delta(x) dx[/tex]
[tex]= f'(x)\delta(x)- f'(0)[/itex].

So [itex]d(\delta(x))/dx[/itex] is the distribution that maps f(x) to [itex]f'(x)\delta(x)- f'(0)[/itex].