Defining Point Price Elasticity Using the Derivative of the Demand Function
For non-linear demand curves, calculus is used to find the price elasticity at a specific point. This is done by taking the limit of the elasticity formula as the change in price (ΔP) approaches zero: . This limit yields a definition of elasticity based on the derivative of the direct demand function, , which is expressed as or . The resulting formula is . Because quantity (Q) is a function of price (), the elasticity formula can be expressed entirely as a function of price by substituting the demand function for Q. Due to the Law of Demand, the derivative of the demand function, , is negative, meaning the price elasticity of demand (ε) is typically a positive value.
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