5. Derivative of a distribution

In this section, we introduce the fundamental notion of the derivative of a distribution. It has the enormous advantage – which explains the importance of distributions for solving differential or partial differential equations – that any distribution is derivable and therefore indefinitely derivable. Like all the above, we define this notion in such a way that it is compatible with the usual notion of derivative of a function, when the distribution is a regular distribution defined by a derivable function.

Let f be a function that is locally integrable. By integrating by parts, we can write, for any function ϕ in