This is the article you're referring to: https://www.johndcook.com/blog/2009/10/25/how-to-differentiate-a-non-differentiable-function/ https://www.johndcook.com

This is the article you're referring to: https://www.johndcook.com/blog/2009/10/25/how-to-differentiate-a-non-differentiable-function/ https://www.johndcook.com/blog/2009/10/25/how-to-differentia... This is the wikipedia page on integration by parts: https://en.wikipedia.org/wiki/Integration_by_parts https://en.wikipedia.org/wiki/Integration_by_parts He's setting u = phi(x), v = f(x). Since he's integrating on the real line, a = -infinity, b = +infinity, so on the right hand side for "uv" you'd have u(+inf)v(+inf) - u(-inf)v(-inf) - and since u (which is phi) is 0 outside of a finite interval, you know these terms are 0. [You obviously can't evaluate functions at +-inf, and you have to take limits to evaluate u(x)v(x) for an improper integral, but you can see the result is the same if u(x) is zero outside of a finite interval]