Marginal Revenue Suppose that the demand function is given...

Marginal Revenue Suppose that the demand function is given by $p=D(q),$ where $q$ is the quantity that consumers demand when the price is $p$ . Show that the marginal revenue is given by
$$R^{\prime}(q)=D(q)+q D^{\prime}(q).$$

Key Concepts

Revenue Function

The revenue function represents the total income received from selling a certain quantity of goods. It is typically defined as the product of the price per unit and the quantity sold. Understanding the revenue function is crucial in economic analysis as it lays the foundation for deriving related concepts such as marginal revenue.

Marginal Revenue

Marginal revenue is defined as the additional revenue that is generated by selling one more unit of a good or service. Mathematically, it is the derivative of the revenue function with respect to quantity. This concept helps businesses determine the optimal level of production, as it helps in understanding how revenue changes with changes in quantity sold.

Demand Function

The demand function is a mathematical relationship that describes how the quantity demanded of a good varies with changes in its price. It is fundamental in economic theory because it captures consumer behavior and market responses. In revenue analysis, the demand function is used to express the price in terms of the quantity, which can then be incorporated into the revenue function.

Product Rule

The product rule is a basic rule of differentiation in calculus used when taking the derivative of a product of two functions. In the context of revenue analysis, it is applied when differentiating the revenue function, which is the product of quantity and price (as given by the demand function). This rule is essential for deriving the marginal revenue formula.

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