5. Derived from a distribution

In this section, we introduce the fundamental concept of the derivative of a distribution. It has the enormous advantage – which explains the importance of distributions for solving differential and partial differential equations – that every distribution is differentiable and therefore infinitely differentiable. As with everything discussed above, we define this concept so that it is compatible with the usual notion of the derivative of a function, when the distribution is a regular distribution defined by a differentiable function.

Let f be a function such that $〈{\left[\ln \left(\left|x\right|\right)\right]}^{\prime },$