the Fréchet derivative is a derivative defined on Banach spaces. Named after Maurice Fréchet, it is commonly used to formalize the concept of the functional derivative used widely in the calculus of variations. Intuitively, it generalizes the idea of linear approximation from functions of one variable to functions on Banach spaces. The Fréchet derivative should be contrasted to the more general Gâteaux derivative which is a generalization of the classical directional derivative.
The Fréchet derivative has applications throughout mathematical analysis, and in particular to the calculus of variations and much of nonlinear analysis and nonlinear functional analysis. It has applications to nonlinear problems throughout the sciences.