Hyper-dual numbers (HDNs) are defined using distinct nilpotent elements. Extending functions from real space to hyper-dual space enables the calculation of high-order derivatives without relying on formulas such as the Leibniz rule or the chain rule. Furthermore, the HDN framework facilitates computing high-order derivatives for implicit functions, such as eigenvalues. This paper presents a comprehensive overview of our research on HDN theory and its application. We explore specific topics such as the HDN-based higher-order derivative formulation, HDN matrix representation, HDN-based numerical differentiation of eigensystems, and its application in hyperelastic–plastic materials. The development of high-accuracy numerical methods using higher-order derivatives and the efficient computing of high-order derivatives are two major benefits of HDN-based numerical differentiation.
