Marginal Cost and Marginal Revenue

The previous chapter built up the machinery of differentiation — limits, derivatives, and the rules for differentiating standard functions. This chapter puts that machinery to work on real business and economic questions: how fast is cost changing as output rises, at what output is profit largest, and how does a firm's decision variable move when it genuinely depends on two quantities together, not one.

Every firm has a total cost function $C(x)$, the total cost of producing $x$ units, and a total revenue function $R(x)$, the total money received from selling $x$ units. The marginal cost is the rate at which total cost changes with output — the (approximate) extra cost of producing one more unit — and is found by differentiating the cost function:

Similarly, the marginal revenue is the rate at which total revenue changes with output:

The average cost is simply total cost spread evenly over output, $AC = \dfrac{C(x)}{x}$, and is a genuinely different quantity from marginal cost — AC tells you the cost per unit averaged over everything made so far, while MC tells you the cost of just the next unit.

Because marginal cost and marginal revenue are both ordinary derivatives, every differentiation rule from the previous chapter — the power rule, sum rule, product/quotient rule — applies directly; no new differentiation technique is needed here, only a new economic reading of what the derivative means.

This marginal-analysis idea — using the derivative of a cost or revenue function to study the next unit rather than the whole batch — is the same calculus-based approach used in the business/economics applications of differentiation taught across Indian commerce boards, including CBSE's own Applied Mathematics elective.