Here is a Fourier series with one hundred terms and plots below.
Here is the first derivative.
The zeros of the first derivative appear to be evenly spaced and known exactly. This made me want to try my numerical differentiation formulae because it is possible to avoid spanning oscillations with the value of h in the formulae. It seems that these calculations would not be useful for derivatives of the unknown functions in differential equations, but they might be useful in the definition of coordinate systems.
The 'modes' b, c, f below stand for backward, central, and forward formulae. The numbers shown below are as follows: First there is the sequence of values produced by the formulae, then there follows the exact value of the derivative, then the sequence of relative errors.