# How to Calculate a Marginal Revenue Derivative

by Ryan Menezes

Economists informally call a product's marginal revenue the increased revenue from a single new sale. This definition is not entirely accurate. That formula finds the increased revenue when the item's quantity increases by one. But a more precise measure of marginal revenue finds the change in revenue from an infinitesimal change in quantity. Calculate this using differential calculus. Marginal revenue is the derivative of the product's revenue with respect to its quantity.

Obtain or estimate a relationship between the item's price and the quantity of units that you sell. This function forms the item's demand curve on a graph. For example, assume that the price of knives is \$20 minus the knives' quantity, or p = 20 - q.

Multiply both sides of the equation by q, the item's quantity, because revenue is the product of price and quantity. Continuing the example from the previous step: p × q = q (20 - q), or revenue is 20q - q^2.

Differentiate the equation with respect to q. To do this, reduce each component's exponent by one ,and multiply the result by the earlier exponent. With this example, differentiating 20q^1 - q^2 gives: [1 × 20 × q^(1 - 1)] + [2 × q^(2 - 1)], or 20 - 2q. This expression represents marginal revenue.

Apply the marginal revenue formula to a sales level. For example, if you want to know the marginal revenue when you sell seven knives: 20 - (2 × 7) = \$6.

#### About the Author

Ryan Menezes is a professional writer and blogger. He has a Bachelor of Science in journalism from Boston University and has written for the American Civil Liberties Union, the marketing firm InSegment and the project management service Assembla. He is also a member of Mensa and the American Parliamentary Debate Association.

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