I'm interested in showing that:
$ \frac{d}{dt}P \; \int_{-\infty}^{\infty} \frac{\phi(x)}{x-t}dt = P \int_{-\infty}^{\infty}\frac{\phi(x)-\phi(t)}{(x-t)^2}dt $
where $\phi(x)$ is a test function (goes to zero at $-\infty$ and $\infty$ fast enough that we don't have to worry about x not going to zero fast enough)
The problem is that when I attempt this:
$ \frac{d}{dt}P \; \int_{-\infty}^{\infty} \frac{\phi(x)}{x-t}dt = \lim_{\epsilon \rightarrow 0} \left\{ -\frac{\phi(t+\epsilon)}{\epsilon}-\frac{\phi(t-\epsilon)}{\epsilon} +\int_{-\infty}^{t-\epsilon}\frac{\phi(x)}{(x-t)^2} + \int_{t+\epsilon}^{\infty}\frac{\phi(x)}{(x-t)^2} \right\} $
But I don't know where to go from here... and I'm not sure I'm on the right track. The $\phi(t)$ term doesn't seem to want to pop out.
Could this be some sort of dirac delta identity I'm missing?
Thanks!