I've seen some [previous questions](https://mathoverflow.net/q/44774) that show that the derivative operator on the set of smooth functions can be given by the Leibniz rule and/or chain rule and some other axioms.


Is there a similar characterization of the derivative $\mathcal{C}^1(\mathbb{R}) \to \mathcal{C}(\mathbb{R})$? Or other interesting cases? (e.g. a differentiation operator on distributions of some suitable type)



Algebraic characterization of real differentiation
Derivative on polynomials can be characterized as a linear map which satisfies
Leibniz rule, zero on constants and $1$ on the identity function.
This extends it uniquely to rational functions. Now in any space where rational functions
are dense, such an operator, if continuous, must be the derivative.


