I believe I have this straight:
The differential (dy) is a good approximation for the change in y (delta-y) when delta-x is small. Therefore, you can use dy to approximate the change in a function (say the cost, revenue, or profit function) that occurs when there is a small change in x (the quantity of a product).
It is not difficult to come up with the equation for dy - simply take the derivative of the function and multiply it times delta-x (which must be small for this to work since dx would 'be' zero if delta-y were to actually equal dy).
In economics, a 'marginal' essentially answers the question "If I were to produce one more or one less, how would that affect my cost, revenue, and/or profit?". Thus, delta-x = 1 when determining the formula for dy.
My question is: Why do this at all? Why not actually use delta-y? You could get a formula for this quite easily and it would be an exact answer, not an approximation. Is there a flaw in my interpretation of a marginal? Am I not seeing the bigger picture somehow?