ECON1003 · Quantitative Methods In Economics
Differentiation
Differentiation is the spine of ECON1003 — it spans both exams and carries the headline application of the whole unit. A derivative is a rate of change, the slope of the tangent, and economics calls that rate marginal. Master the rules (power, constant, sum, product, quotient, chain, plus the aˣ and log derivatives on the sheet), then read what the first and second derivatives say about a curve's shape — increasing/decreasing, concave up/down, turning points and inflection points. The payoff is optimisation: given demand and cost, find the profit-maximising output where MR = MC, and confirm it with the second-derivative test. The limit definition is shown for understanding, but the lecturer states he will not ask you to differentiate from the limit.
What this chapter covers
- 014.1 The derivative and the differentiation rules
- 02Product, quotient and chain rules
- 034.2 Higher derivatives, concavity and the second-derivative test
- 044.3 Turning points and points of inflection
- 054.4 Economic applications: marginal functions
- 06Profit maximisation where MR = MC
Worked example: differentiate, then optimise (MR = MC)
- +1(a) Set up profit. TR = PQ = (120 − 3Q)Q = 120Q − 3Q²; π = TR − TC = 120Q − 3Q² − 30Q − 100 = 90Q − 3Q² − 100.
- +1Differentiate and set to zero. dπ/dQ = 90 − 6Q = 0.
- +1Solve. 6Q = 90 ⇒ Q* = 15. (Equivalently MR = 120 − 6Q equals MC = 30 at Q = 15.)
- +1(b) Second-derivative test. d²π/dQ² = −6 < 0 ⇒ concave ⇒ a maximum.
- +1(c) Price. P* = 120 − 3(15) = $75.
- +1Maximum profit. π = 90(15) − 3(15)² − 100 = 1350 − 675 − 100 = $575.
Key terms
- Derivative
- The instantaneous rate of change of a function, the slope of its tangent: f′(x). In economics it is the 'marginal' quantity — marginal cost is dTC/dQ, marginal revenue is dTR/dQ.
- Chain rule
- The rule for differentiating a function of a function: with u = g(x), dy/dx = (dy/du)(du/dx). Needed whenever something is raised to a power or sits inside another function.
- Second-derivative test
- A test of a turning point's type: at a stationary point (f′ = 0), f″ < 0 means a maximum (concave), f″ > 0 means a minimum (convex). The step students forget that costs the last optimisation mark.
- Inflection point
- A point where concavity changes sign (f″ = 0 and changes sign). The curve switches from concave up to concave down or vice versa; it is not a turning point of f.
- MR = MC
- The profit-maximising condition. Setting dπ/dQ = 0 gives dTR/dQ = dTC/dQ, i.e. marginal revenue equals marginal cost. Confirm with the second-derivative test that it is a maximum, not a minimum.
Differentiation FAQ
Do I have to use the limit definition of the derivative?
No. The limit f′(x) = limₖ→₀[f(x+k) − f(x)]/k is shown so you understand what a derivative is, but the lecturer states he will not ask you to differentiate from the limit. In practice you always use the rules — power, constant, sum, product, quotient and chain — not the limit.
Which differentiation rules must I memorise versus which are on the sheet?
The formula sheet typically gives you the aˣ and logₐ derivatives and the quotient rule; you must know the power, constant, sum, chain and product rules cold. A useful habit: rewrite roots and fractions as powers first (√x → x^0.5, 1/x → x^−1) so the power rule n·x^(n−1) just works.
How do I find the profit-maximising output?
Set up profit π = TR − TC with TR = P×Q, differentiate to get dπ/dQ, set it to zero and solve for Q. That first-order condition is exactly MR = MC. Then always run the second-derivative test: d²π/dQ² < 0 confirms a maximum. Quoting Q* without the sign check drops the last mark.
Exam move
Differentiation carries the most marks and spans both exams, so weight your time here. Get the rules to reflex speed, and always rewrite roots and fractions as powers before applying the power rule. The exam favourite is the MR = MC profit-maximisation chain: set up TR = P×Q, differentiate profit, set dπ/dQ = 0, solve, then ALWAYS confirm with the second-derivative test — the sign check is a free mark students routinely forget. Show every line; partial credit on the rules and the FOC is real even if the final number slips.
