 If formula_1 is a test function, we can use integration by parts to see that

where the last equality follows from the fact that formula_1 has compact support, so is zero outside of a bounded set.This suggests that if formula_17 is a "distribution", we should define its derivative formula_18 by

It turns out that this is the proper definition; it extends the ordinary definition of derivative, every distribution becomes infinitely differentiable and the usual properties of derivatives hold.Example