Show that [tex]x \frac{d(\delta (x))}{dx} = -\delta (x)[/tex](adsbygoogle = window.adsbygoogle || []).push({});

where [itex]\delta (x)[/itex] is a Dirac delta function.

My work:

Let f(x) be a arbitrary function. Using integration by parts:

[tex]\int_{-\infty}^{+\infty}f(x)\left (x \frac{d(\delta (x))}{dx}\right)dx = xf(x)\delta (x)\vert _{-\infty}^{+\infty} - \int_{-\infty}^{+\infty}d\left (\frac{xf(x) \delta (x)}{dx}\right)dx[/tex]

The first term is zero, since [itex]\delta (x) = 0 [/itex]

at [itex]-\infty, +\infty[/itex].

How is the second term evaluated?

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