A differentiator FIR filter produces approximately the derivative of the input signal over a certain range of frequencies..
To design an differentiator FIR filter, Typically, a low-pass filter is designed, although the gain at zero frequency must be zero, so three-bands might be used for the low-pass filter, with the first band having a relatively small transition region.
The resulting filter is antisymmetric, and thus has linear phase response with frequency.
For a filter with length N:
h(k) = -h(N-k)
k = 0, 1, 2, ... N - 1
For an odd-length filter, the center coefficient is zero.