Way to think about weak deriviate

Question

Something hit me when I read the definiton of weak derivite. Would it be right to think about the weak deriviate in terms of distributons, i.e that the distribution $\int f \phi$ induced by f in $ L^{1} $ changes to $\int f' \phi$ (up to sign) when we "take the weak of $f$ deriviate". Where $f'$ is the function in the definition of weak derivative. Im woundering if I got the main idea behind something here either distrubutions or weak deriviates. I dont know if distributions was defined to handle weak deriviates or if they originate from somewhere else.

Answer 1

From comment;

That's pretty much exactly the point. When the distributional derivative is representable as an Lp function, that is the weak derivative.