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This document discusses Gateaux differentials and Frechet derivatives, which are generalizations of derivatives to vector spaces. It begins by introducing the concept of the Gateaux differential, which generalizes the directional derivative. It then provides examples of calculating the Gateaux differential for linear and quadratic functions, the exponential function, and functions involving integration. Finally, it establishes basic rules for computing Gateaux differentials, such as the sum rule, product ru
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