Why we study distributional derivative?

Let $\Omega\subset \Bbb{R}^n$ be any open set.

$D(\Omega)=C_c^{\infty}(\Omega) $ : Linear space of test functions i.e smooth functions with compact support.

$D'(\Omega) $: Continuous dual of $D(\Omega) $

For $f\in D'(\Omega) $ we define distributional derivative of $f$ , $D^{\alpha}f$ or $\partial^{\alpha}f$ by

$$\langle\partial^{\alpha}f,\varphi\rangle=(-1)^{|\alpha|}\langle f,\partial^{\alpha}\varphi\rangle$$

There are locally integrable function which is not differentiable in classical sense but the regular distribution generated by the locally integrable function possess distributional derivative.

What is the intuition behind distributional derivative and why distributional derivative is useful?

Can you explain some application where we need some sort of differentiation but classical differentiation is no longer useful?