Distributional derivative of the logarithmic modulus ($\frac{d}{dx}\log |x|=\mathrm{pv}(1/x)$)

The locally integrable function $\log |x|$ has distributional derivative equal to the principal-value reciprocal distribution. Symmetric integration by parts makes the boundary term $\log \epsilon [\phi (\epsilon )-\phi (-\epsilon )]$ tend to zero.