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, providing a linear approximation of the function at a given point. This derivative is defined in terms of limits, similar to the traditional derivative, but accommodates more complex spaces. In mathematical terms, a function is Fréchet differentiable at a point if there exists a bounded linear operator that approximates the function's change near that point. This concept is crucial in functional analysis and has applications in optimization and differential equations, where understanding the behavior of functions in infinite-dimensional spaces is essential.