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Differentiation is a crucial mathematical tool used to determine the rate at which a variable changes. In the context of economics, particularly in demand analysis, differentiation helps find how demand for a product changes as its price changes. To differentiate a demand function like \[ D(p) = 300 - p^2 \] with respect to price \( p \), you calculate its derivative \( D'(p) \). This derivative helps identify the sensitivity of demand to changes in price. For our example, the derivative is \[ D'(p) = -2p \].This negative coefficient of \( p \) implies that the demand decreases as the price increases, a common economic behavior. Differentiation thus gives us a mathematical view of how demand responds to price changes.

A demand function is an equation that illustrates the relationship between the quantity demanded of a good and its price. Simply put, it shows how much of a product consumers are willing to buy at varying price levels. In the provided example, the demand function is given by:\[ D(p) = 300 - p^2 \]The function suggests that as the price \( p \) increases, the demand \( D(p) \) decreases, which mirrors the typical law of demand in economics. Understanding the demand function allows businesses and economists to predict consumer behavior and optimize pricing strategies to maximize revenue. It provides insights into how pricing changes can impact sales volume, which is invaluable for making informed business decisions.

The concept of elasticity of demand describes how sensitively the quantity demanded responds to changes in price. Here's how you can understand it:- **Elastic Demand** occurs if \( |E(p)| > 1 \), meaning consumers are highly responsive to price changes.- **Inelastic Demand** implies \( |E(p)| < 1 \), indicating consumers are less responsive to price changes.- **Unit-Elastic Demand** is when \( |E(p)| = 1 \), meaning the percentage change in quantity demanded is exactly equal to the percentage change in price.In our specific example, where the elasticity was calculated to be \( E(10) = -1 \), the demand is deemed unit-elastic at the price \( p=10 \). This means that a 1% change in price results in exactly a 1% change in the quantity demanded. Understanding the elasticity type helps in setting effective pricing policies and anticipating consumer reactions.

Unit elasticity denotes a balanced response of quantity demanded in relation to price changes. Specifically, it occurs when the absolute value of the elasticity of demand equals one, \[ |E(p)| = 1 \].In this scenario, percentage changes in price result in equivalent percentage changes in quantity demanded. For instance, if price increases by 5%, demand decreases by 5%, balancing the effect on total revenue. In the given solution's context, the calculation showed \[ E(10) = -1 \], indicating unit elasticity at \( p = 10 \). This signifies a careful equilibrium between price adjustments and demand shifts, ensuring revenue remains relatively stable despite price fluctuations. Recognizing unit elasticity is important for businesses aiming to maintain stable revenue when experimenting with pricing strategies.