What is the intuition behind distributional derivative and why distributional derivative is useful? 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?
 A: 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.
A: The intuition basically comes from integration by parts.
Ignoring details, suppose $f$ and $g$ are really nice, and suppose we want to study
$$ \int_a^b \frac{df}{dx}(x) g(x) dx.$$
Integration by parts tells us that this is the same thing as studying
$$ \int_a^b \frac{df}{dx}(x) g(x) dx = f(x)g(x) |_{x=a}^b - \int_a^b f(x)\frac{dg}{dx}(x)dx.$$
If everything works out as we want it to, then what we end up getting is
$$ \int_a^b \frac{df}{dx}(x) g(x) dx = - \int_a^b f(x)\frac{dg}{dx}(x)dx$$
which should look very familiar. For example, if $b = \infty$ and $a = -\infty$ and these are compactly supported smooth functions, then this is what will happen.
These types of things are necessary in differential equations -- see, for example, this: http://web.math.ucsb.edu/~grigoryan/124A/lecs/lec10.pdf
A: Complex numbers are often useful to prove theorems only involving real numbers (e.g., every polynomial over R can be factorised into a product of degree 1 and degree 2 polynomials over R).
In the same way, it's not surprising that more general objects than ordinary functions, which can ALWAYS be differentiated, can be useful in PDEs, etc.
That's the whole purpose of distributions: to have a class of objects extending ordinary functions, which can always be differentiated, and agree with the ordinary derivative whenever they are smooth functions.
For example, look up elliptic regularity. Sometimes you can prove that a PDE has a distribution solution, more easily than an ordinary function. Then prove a SEPARATE theorem that the distribution solution in fact IS just an ordinary function.
Look up things like the fundamental solution and Green's functions for PDEs.
E.g., to solve $Lu=f$ for some linear differential operator $L$, first solve $Lk = \delta$ for the Dirac delta function $\delta$. Then, formally, using convolution:
$L(k*f) = Lk * f = \delta * f = f$.
Thus we've found a general formula $u=k*f$ to generate solutions. Sometimes, as in elliptic regularity, we can use properties of $k$ to prove theorems about the smoothness of $u$, given smoothness conditions on $f$, etc.
Distribution theory can sometimes make such arguments rigorous.
Of course, in any given special case, it might be that results can be proved directly and easily without the general theory; but this is true of any general theory.
Also very important is the Fourier transform, which works on a subspace of distributions called tempered distributions. Again, this generalises the ordinary Fourier transform for nice functions.
It should also be remembered that physicists and engineers were already doing calculations like this before distribution theory. So it's nice to have a fully rigorous general theory to lay more solid foundations (although, some details are rather advanced mathematically. E.g., you need the Lebesgue differentiation theorem to prove that ordinary locally integrable functions define a unique distribution, but this fact is sometimes unmentioned. Also, Banach spaces and convergence in ordinary metric spaces are not enough to define distributions rigorously).
