The Gateaux derivative is a concept in functional analysis that generalizes the idea of differentiation to functions defined on infinite-dimensional spaces. It is particularly useful in the study of Banach spaces and Hilbert spaces, where traditional derivatives may not apply. The Gateaux derivative measures the rate of change of a function in a specific direction, allowing for a more flexible approach to differentiation. To compute the Gateaux derivative, one evaluates the limit of the difference quotient as a small perturbation is applied in a given direction. This derivative is defined for a function f at a point x and in the direction of a vector h . If the limit exists, it provides valuable information about the behavior of the function near that point.