other stencil configurations and derivative orders, the Finite Difference Coefficients Calculator is a tool that can be used to generate derivative approximation methods for any stencil with any derivative order (provided a solution exists). Higher derivatives Using Newton's difference quotient, the following can be shown (for n>0): Complex-variable methods The classical finite-difference approximations for numerical differentiation are ill-conditioned. However, if is a holomorphic function, real-valued on the real line, which can be evaluated at points in the complex plane near , then there are stable methods. For example, the first derivative can be calculated by the complex-step derivative formula: The recommended step size to obtain accurate derivatives for a range of conditions is . This formula can be obtained by Taylor series expansion: The complex-step derivative formula is only valid for calculating first-order derivatives. A generalization of the above for calculating derivatives of any order employs multicomplex numbers, resulting in multicomplex derivatives. where the denote the multicomplex imaginary units; . The operator extracts the th component of a multicomplex number of level , e.g., extracts the real component and extracts the last, “most imaginary” component. The method can be applied to mixed derivatives, e.g. for a second-order derivative A C++ implementation of multicomplex arithmetics is available. In general, derivatives of any order can be calculated using Cauchy's integral formula: where the integration is done numerically. Using complex variables for numerical differentiation was started by Lyness and Moler in 1967. Their algorithm is applicable to higher-order derivatives. A method based on numerical inversion of a complex Laplace transform was developed by Abate and Dubner. An algorithm that can be used without requiring knowledge about the method or the character of the function was developed by Fornberg. Differential quadrature Differential quadrature is the approximation of derivatives by using weighted sums of function values. Differential quadrature is of practical interest because its allows one to compute derivatives from noisy data. The name is in analogy with quadrature, meaning numerical integration, where weighted sums are used in methods such as Simpson's method or the Trapezoidal rule. There are various methods for determining the weight coefficients, for example, the Savitzky–Golay filter. Differential quadrature is used to solve partial differential equations. There are further methods for computing derivatives from noisy data. See also List of numerical-analysis software References External links Numerical Differentiation from wolfram.com Numerical Differentiation Resources: Textbook notes, PPT, Worksheets, Audiovisual YouTube Lectures at Numerical