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?
Best Answer
And as to "why would we want this?": well, sometimes we might want to integrate by parts, or differentiate, or do other standard calculus-y stuff, in the interior of a larger computation or proof, but do not know whether the thing we want to differentiate is (classically) differentiable.
One great aspect of the distributional point of view is that it extends classical differentiation consistently with essentially all other calculus operations. So we can proceed without worrying about classical differentiability.
Similarly, we can take Fourier transforms of many things (tempered distributions) without worrying about whether they're in $L^1$ or $L^2$. Consistently with expected properties of Fourier transform.