A weak derivative is a generalization of the concept of a derivative that allows for the differentiation of functions that may not be smooth or well-behaved. Unlike traditional derivatives, which require functions to be differentiable at every point, weak derivatives can be defined for functions in the context of distributions or generalized functions. This approach is particularly useful in the study of partial differential equations and functional analysis. In mathematical terms, a weak derivative is defined through integration by parts. If a function u is in a certain space, its weak derivative u' exists if, for all test functions ϕ in a suitable space, the integral of u times the derivative of ϕ equals the integral of u' times ϕ. This concept allows for the treatment of functions that are not differentiable in the classical sense, expanding the scope of analysis in various mathematical fields.