Closed-form expressions for the finite difference approximations of first and higher derivatives based on Taylor series

Numerical differentiation formulas based on interpolating polynomials, operators and lozenge diagrams can be simplified to one of the finite difference approximations based on Taylor series. In this paper, we have presented closed-form expressions of these approximations of arbitrary order for first and higher derivatives. A comparison of the three types of approximations is given with an ideal digital differentiator by comparing their frequency responses. The comparison reveals that the central difference approximations can be used as digital differentiators, because they do not introduce any phase distortion and their amplitude response is closer to that of an ideal differentiator. It is also observed that central difference approximations are in fact the same as maximally flat digital differentiators. In the appendix, a computer program, written in MATHEMATICA is presented, which can give the approximation of any order to the derivative of a function at a certain mesh point.