Introduction

Numerical differentiation refers to approximating the value of a derivative of a function (or a higher order derivative) numerically:

If is known at a number of data points, can we deduce a reliable value for at one of the data points?

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Like interpolation, numerical differentiation is a basic numerical task, often needed in more complex numerical simulations.

We will be deriving formulae starting from either a Taylor series expansion or from the result of Theorem 10: