Frequency response of numerical differentiation by polynomial interpolation / finite difference
Polynomials are not integrable, hence their Fourier interpretation is complicated. However, locally, there are a few works. Under least-squares, polynomial interpolation's linearity works well with frequency analysis.

The class of Savitsky-Golay filters performs least-squares polynomial interpolation, and yields different derivative orders. As they seemed to be little known in signal processing, R. W. Schafer dedicated them What Is a Savitzky-Golay Filter? (public version) in the Lecture Notes section of IEEE Signal Processing Magazine in 2011. There, he especially covers their properties, including frequency responses. Analysis was also performed in Smoothing by Savitzky-Golay and Legendre filters by P. O. Persson and G. Strang in 2003. One can combine derivative accuracy and noise shaping or frequency response fitting.

More recently, there were derivations with IIR filters and fractional differentiation, for instance Maximally Flat IIR Smoothers With Repeated Poles and a Prescribed Delay, and even non-linear versions.

In image processing, there has been a lot of works as well, I can dig into my reference database if needed.