The Fréchet derivative is a generalization of the concept of a derivative in calculus, applied to functions between Banach spaces. It measures how a function changes as its input varies, capturing the idea of linear approximation. If a function is Fréchet differentiable at a point, it can be approximated by a linear map near that point. In mathematical terms, the Fréchet derivative at a point is a linear operator that best approximates the change in the function. This concept is crucial in functional analysis and is used in various fields, including optimization and differential equations, to analyze the behavior of complex functions.