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 $n$th Sobolev space: closure of test functions under the $n$th 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 $\frac{d}{dx}$ extends by continuity to map $n$th 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 $n$th Sobolev implies $u$ is in the $n+\operatorname{deg}(D)$ 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...
