Presentation, error analysis and numerical experiments on a group of 1-step-ahead numerical differentiation formulas

In order to achieve higher computational precision in approximating the first-order derivative of the target point, the 1-step-ahead numerical differentiation formulas are presented. These formulas greatly remedy some intrinsic weaknesses of the backward numerical differentiation formulas, and overcome the limitation of the central numerical differentiation formulas. In addition, a group of formulas are proposed to obtain the optimal step length. Moreover, the error analysis of the 1-step-ahead numerical differentiation formulas and the backward numerical differentiation formulas is further investigated. Numerical studies show that the proposed optimal step-length formulas are effective, and the performance of the 1-step-ahead numerical differentiation formulas is much better than that of the backward numerical differentiation formulas in the first-order derivative approximation.