Show that the derivative of P.V. $(\frac{1}{x})$ is $f^\prime[\delta] = - \ \text{lim}_{\epsilon \rightarrow 0} \int_{|x|\geq \epsilon} \frac{\phi(x) - \phi(0)}{x^2} dx$
To prove this I have written $\langle (P.V. (\frac{1}{x}))^\prime, \phi(x)\rangle = \langle P.V. (\frac{1}{x}), \phi^\prime (x) \rangle = \int P.V. (\frac{1}{x}) \phi^\prime (x) \ dx$.
I also know that P.V. $(\frac{1}{x}) = \int_{|x|> \epsilon} \frac{\phi(x)}{x} dx$
But how to proceed further? Please help me. I have no idea.