Numerical differentiation is a technique used to approximate the derivative of a function based on discrete data points, rather than relying on analytical methods. This approach is particularly useful when dealing with functions that are difficult to differentiate analytically or when only a set of sampled points is available, such as experimental or observational data.
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Numerical differentiation estimates derivatives from finitely many function values, often by differentiating an interpolation polynomial or using a finite difference. Its error combines truncation error from the formula with errors already present in the sampled values and additional roundoff error. Decreasing the sample spacing can decrease truncation error while increasing the effect of uncertain values. This tradeoff is explained by the ill-posedness of numerical differentiation when only an absolute value-error bound is available.