There is a certain refinement of the question which turns out to have useful, interesting content, namely, talking about (L^2$L^2$) Sobolev spaces, both local and global. For simplicity, on the real line, the 0th Sobolev space is just L^2$L^2$. For positive integer n$n$, there are three characterizations of the nth$n$th Sobolev space: closure of test functions under the nth$n$th Sobolev norm-squared:
|f|^2_n = |f|^2 + |f'|^2 + |f''|^2 + ...+|f^(n)|^2$$|f|^2_n = |f|^2 + |f'|^2 + |f''|^2 + ...+|f^{(n)}|^2$$
closure of smooth functions whose n derivatives are in L^2$L^2$, under the same norm, and the collection of distributions in L^2$L^2$, whose n derivatives are in L^2$L^2$.
The d/dx$\frac{d}{dx}$ extends by continuity to map nth$n$th to (n-1$n-1$)th Sobolev space, and is "L^2"$L^2$ 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$n-1$)-times continuously differentiable functions. (In dimension N$N$, the discrepancy is N/2 + epsilon$N/2 + \epsilon$.)
In higher dimensions, "elliptic regularity" is the assertion (proven decades ago) that operators D$D$ such as the Laplacian have the property that Du=f$Du=f$ with f$f$ in nth$n$th Sobolev implies u$u$ is in the n+deg(D)$n+\operatorname{deg}(D)$ Sobolev space. Part of the technical point here is that the proof really proves something about L^2$L^2$ differentiability, not classical.
In fact, I would claim that this circle of ideas deserves to be part of every mathematician's worldview...
