Marginal cost is defined as "the change in the total cost that arises when the quantity produced is incremented by one unit." And given a total cost function $C(q)$ that's differentiable, the marginal cost is the derivative, $C'(q)$. But if I were given $C$ and asked the cost that arises when the quantity produced is increased from 2 to 3, I would simply calculate $C(3)-C(2)$; no need to bring calculus into the picture. In general, $ C(3)-C(2) \neq C'(2)$. For example, if $C(q) = q^2$, then $C(3)-C(2) = 5$, but $C'(2) = 4$.
Thus my question is: Why is the derivative used to represent marginal cost instead of the difference?
Note: I thought this question must've been what's being asked here, but evidently not; there what's being asked is (essentially) why $C'(3) \neq C(3)-C(2)$.