Distributional Derivative

A distributional derivative extends the concept of derivatives to functions that may be irregular or not differentiable in the traditional sense, such as functions with jumps or spikes. Instead of relying on pointwise limits, it uses the framework of distributions (generalized functions) to capture the notion of "rate of change." This approach involves integrating against test functions and applying integration by parts, allowing us to differentiate a broader class of functions. Distributional derivatives are essential in solving differential equations with discontinuities and modeling phenomena with abrupt changes.