Applications of Derivatives (Commerce Contexts)
This domain applies derivatives to cost, revenue, profit and demand: marginal functions, elasticity of demand, and optimisation. Expect questions that give a total function and ask for a marginal value at a stated output, or ask for the output maximising profit or revenue.
Marginal and average functions
If C(x), R(x) and P(x) are total cost, revenue and profit, then MC = dC/dx, MR = dR/dx and MP = dP/dx. Marginal values approximate the change from producing one extra unit. If p is the price per unit and x the quantity, R = px, so MR = p + x(dp/dx). Average cost is AC = C/x, which is not MC.
Elasticity of demand
η = −(p/x)(dx/dp). Demand is elastic when |η| > 1, inelastic when |η| < 1 and unit elastic when |η| = 1.
Maxima and minima
Solve f′(x) = 0, then test: f″(x) < 0 gives a maximum, f″(x) > 0 a minimum. Profit is maximised where MR = MC with MC rising faster than MR, equivalently where dP/dx = 0 and P″ < 0.
Worked example
A firm has R(x) = 50x − x² and C(x) = 10x + 100. Then P(x) = 40x − x² − 100, so P′(x) = 40 − 2x = 0 gives x = 20. Since P″(x) = −2 < 0, x = 20 is a maximum and P(20) = 800 − 400 − 100 = 300. Check the condition: MR = 50 − 2(20) = 10, and MC = 10, so MR = MC at the optimum.
Traps
- Confusing marginal cost with average cost.
- Treating price p as constant when differentiating R = px; MR equals p only if price is fixed.
- Accepting any stationary point as a maximum without the second-derivative test.
- Dropping the minus sign in elasticity and then mislabelling elastic and inelastic ranges.