When someone wants to identify the derivative of a distribution $T\in \mathcal{D}'(\mathbb{R})$, we usually write, for $\varphi\in\mathcal{D}(\mathbb{R})$ , $$\langle T',\varphi\rangle = -\langle T,\varphi'\rangle $$ and then do an integration by parts to get something like $\langle T',\varphi\rangle=\langle U,\varphi\rangle$ and then conclude that $T'=U$. But one may argue that the exercise makes no sense, since the derivative of $T$ is $\varphi \mapsto -\langle T,\varphi'\rangle$ and there is nothing to do. Especially when the expression, $U$, that we get after integration(s) by parts is as complicated as $\varphi \mapsto -\langle T,\varphi'\rangle$.
Am I right ? Or do I miss something ?
As is often the case in mathematics, two questions (or more) are mixed together: asking what is the definition of a thing, and asking for a useful/simpler recharacterization. As the question notes, the definition of the derivative of a distribution is clear, so "we are done" in that regard. But often, when the distribution is given by integration against a function, for example, it is useful to see that/whether its derivative is also given by integration against a function... or not-quite.
That is, in practice, there is reason to want to see how close a distribution is to being a locally integrable function, or being continuous, etc. Some of these issues can be systematized by various forms of (Levi-) Sobolev space considerations.