In recent decades an effort has been made to generalize results from classical analysis in "smooth settings" to "non-smooth" ones. This includes the modelling of physical phenomena such as heat flow or particle movement. Furthermore, efforts have been made to introduce analogues of derivatives and differential operators in these contexts. This talk opens with presenting one of the attributes such a derivative is aiming to fulfill, namely an analogue of the Poincare inequality. Furthermore, there is a broader functional analysis context to the problem which will be introduced and discussed.