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Finite differences have been widely used in mathematical theory as well as inscientific and engineering computations. These concepts are constantlymentioned in calculus. Most frequently-used difference formulas provideexcellent approximations to various derivative functions, including those usedin modeling important physical processes on uniform grids. However, ourresearch reveals that difference approximations on uniform grids cannot beapplied blindly on nonuniform grids, nor can difference formulas to formconsistent approximations to second derivatives. At best, they may loseaccuracy; at worst they are inconsistent. Detailed consistency and erroranalysis, together with simulated examples, will be given.

An Exploration of the Approximation of Derivative Functions via Finite Differences