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  • Price Elasticity of Demand

  • Law of Demand

  • Direct Demand Function: Quantity as a Function of Price (Q = D(P))

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: ε=lim⁡ΔP→0−PQΔQΔPε = \lim_{\Delta P \to 0} -\frac{P}{Q} \frac{\Delta Q}{\Delta P}. This limit yields a definition of elasticity based on the derivative of the direct demand function, Q=g(P)Q = g(P), which is expressed as dQ/dPdQ/dP or g′(P)g'(P). The resulting formula is ε=−PQdQdPε = -\frac{P}{Q} \frac{dQ}{dP}. Because quantity (Q) is a function of price (Q=g(P)Q=g(P)), 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, dQ/dPdQ/dP, is negative, meaning the price elasticity of demand (ε) is typically a positive value.

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