There is a certain refinement of the question which turns out to have useful, interesting content, namely, talking about (L^2) Sobolev spaces, both local and global. For simplicity, on the real line, the 0th Sobolev space is just L^2. For positive integer n, there are three characterizations of the nth Sobolev space: closure of test functions under the nth Sobolev norm-squared:

|f|^2_n = |f|^2 + |f'|^2 + |f''|^2 + ...+|f^(n)|^2

closure of smooth functions whose n derivatives are in L^2, under the same norm, and the collection of distributions in L^2, whose n derivatives are in L^2.

The d/dx extends by continuity to map nth to (n-1)th Sobolev space, and is "L^2 differentiation". It is not classical.

And/but this is not just a bunch of definitions: Sobolev's imbedding theorem shows that the nth Sobolev space is inside (n-1)-times continuously differentiable functions. (In dimension N, the discrepancy is N/2 + epsilon.)

In higher dimensions, "elliptic regularity" is the assertion (proven decades ago) that operators D such as the Laplacian have the property that Du=f with f in nth Sobolev implies u is in the n+deg(D) Sobolev space. Part of the technical point here is that the proof really proves something about L^2 differentiability, not classical.

In fact, I would claim that this circle of ideas deserves to be part of every mathematician's worldview...