Well, this is basically the same answer as the one by Alexandre Eremenko, but here goes: The particular form of the definition of derivative is crucial for partial differential equations. Using weaker notions than classical derivatives leads to a whole new theory of solutions. While this is (part of) what the answer about distributions is about, the particular case of weak derivatives due to Sobolev predates distributions and is of prime importance to the field of PDEs. Also, weak derivatives allow for a rich theory of function spaces (Banach and Hilbert ones).