derivativesstructures

Derivative structures are a computational concept used to represent a function together with its derivatives up to a chosen order with respect to several variables. They enable automatic differentiation and the construction of multivariate Taylor expansions in a compact, manipulable form.

A derivative structure DS encodes the function value and all partial derivatives up to a specified order at a nominal point. For a function f: R^n -> R, a DS of order d stores the collection of partial derivatives ∂^k f / ∂x^k for all multi-indices k with total degree |k| ≤ d. The information is often organized as a structured array or polynomial-like object so that compositions and arithmetic propagate derivatives automatically.

Arithmetic on DS objects is defined so that applying functions and composing DS objects yields a DS representing the value and derivatives of the resulting function. This relies on the multivariate chain rule and related combinatorial rules, making forward-mode automatic differentiation efficient for computing gradients, Hessians, and higher-order derivatives.

The size of a derivative structure grows with the number of variables n and the order d; the total number of stored derivatives is binomial(n + d, d). This growth mirrors the complexity of multivariate Taylor series and reflects the power and cost of higher-order differentiation.

Applications include automatic differentiation for optimization, sensitivity analysis, and the evaluation of Taylor expansions in physics, engineering, and financial math. DS concepts are implemented in numerical libraries under terms like DerivativeStructure, and they relate to jet spaces and symbolic differentiation while offering exact derivative propagation within floating-point precision. Example: a single variable, order 1 DS stores f and f’; order 2 also stores f'' alongside. See also automatic differentiation, Taylor series, and jet theory.