a.k.a. spectral resampling/differentiation/integration.
Rick Lyons, How to Interpolate in the Time-Domain by Zero-Padding in the Frequency Domain. Also more classic Rick Lyons: FFT Interpolation Based on FFT Samples: A Detective Story With a Surprise Ending.
Steven Johnson’s Notes on Fourier-based interpolation) is all I really need here; it points out a couple of subtleties about DTFT-based differentiation of functions.