Applications of Differentiation
Applications of Differentiation connects calculus to business decisions: cost, revenue, profit, marginal analysis, elasticity of demand and optimisation. The exam expects you to differentiate correctly and interpret the derivative economically.
Core rules
- Marginal cost MC = dC/dx, marginal revenue MR = dR/dx, marginal profit = dP/dx.
- Average cost AC = C/x; average revenue AR = R/x. Do not confuse an average with a marginal value.
- Profit P(x) = R(x) − C(x). Maximum or minimum profit occurs where P′(x) = 0; use P″(x) < 0 for a maximum and P″(x) > 0 for a minimum.
- At profit maximum, MR = MC, provided the second-order condition holds.
- Price elasticity of demand is η = −(p/x)(dx/dp). Demand is elastic if |η| > 1, inelastic if |η| < 1, and unit elastic if |η| = 1.
Worked example
A firm has C(x) = x² + 10x + 50 and R(x) = 50x − x². P(x) = (50x − x²) − (x² + 10x + 50) = 40x − 2x² − 50. P′(x) = 40 − 4x. Set 40 − 4x = 0, giving x = 10. P″(x) = −4 < 0, so x = 10 gives a maximum. P(10) = 400 − 200 − 50 = 150.
Traps
Maximising revenue instead of profit; forgetting the fixed cost in the profit function; giving the output x when the question asks for maximum profit; ignoring the second-derivative test; and dropping the negative sign in elasticity, which reverses the elastic/inelastic conclusion.