Given a cost function c(x), and knowing the cost for some large enough x, say, 500 units, we can find the incremental cost of producing the 501st unit, by computing c'(500). And then we can also find the incremental cost of producing the 502nd unit by computing c'(501). So, with this information, we know the incremental costs for producing an extra 2 units, from 500 to 502.
My students weren't satisfied at all with this example, because many of them were perplexed about why I couldn't just compute c(502) - c(500) to get my answer. I was stumped and told them that I don't think they are wrong, but at the moment, I did not have a better answer for them to convince them that the marginal cost function + using derivatives is a good thing.
What can I tell them to highlight the usefulness of the marginal cost function and derivatives?