Differentiation and marginal analysis
The derivative dy/dx is the rate of change of y as x changes. In business, the derivative of total cost or revenue is marginal cost or revenue.
Rules of differentiation
| Rule | Formula | Example |
|---|---|---|
| Constant | d/dx (k) = 0 | d/dx (5,000) = 0 |
| Power | d/dx (xⁿ) = nxⁿ⁻¹ | d/dx (x⁴) = 4x³ |
| Constant multiple | d/dx (kf) = k f′ | d/dx (7x²) = 14x |
| Sum / difference | (f ± g)′ = f′ ± g′ | d/dx (x³ − 2x) = 3x² − 2 |
| Product | (uv)′ = u′v + uv′ | d/dx [x(x + 1)] = 2x + 1 |
| Quotient | (u/v)′ = (u′v − uv′) ÷ v² | d/dx [x/(x + 1)] = 1/(x + 1)² |
| Chain | d/dx f(g(x)) = f′(g(x)) × g′(x) | d/dx (2x + 3)⁵ = 10(2x + 3)⁴ |
| Exponential | d/dx (eˣ) = eˣ | d/dx (3eˣ) = 3eˣ |
| Logarithm (ln = log to base e) | d/dx (ln x) = 1/x | d/dx (5 ln x) = 5/x |
Business meanings
The power rule
Differentiate y = 4x³ − 5x² + 7x − 9 and find the slope at x = 2.
- Differentiate term by term: 4 × 3x² − 5 × 2x + 7 − 0.
- dy/dx = 12x² − 10x + 7.
- At x = 2: 12(4) − 10(2) + 7 = 48 − 20 + 7 = 35.
dy/dx = 12x² − 10x + 7; slope 35 at x = 2.
Product, chain and quotient rules
Differentiate (a) y = (2x + 1)(x² − 3), (b) y = (3x² + 1)⁴, (c) y = x/(x + 1).
- (a) u = 2x + 1, v = x² − 3, u′ = 2, v′ = 2x.
- dy/dx = 2(x² − 3) + (2x + 1)(2x) = 2x² − 6 + 4x² + 2x = 6x² + 2x − 6.
- Check: y = 2x³ + x² − 6x − 3, so dy/dx = 6x² + 2x − 6. ✓
- (b) Chain: outside power 4, inside derivative 6x: dy/dx = 4(3x² + 1)³ × 6x = 24x(3x² + 1)³.
- (c) Quotient: u = x, v = x + 1: dy/dx = [1 × (x + 1) − x × 1] ÷ (x + 1)² = 1/(x + 1)².
(a) 6x² + 2x − 6 (b) 24x(3x² + 1)³ (c) 1/(x + 1)²
Marginal cost and marginal revenue
A Biyagama factory has C(x) = 2,000 + 40x + 0.05x² and R(x) = 150x − 0.2x² (Rs., x units a day). Find MC and MR at x = 200 and interpret.
- MC = C′(x) = 40 + 0.1x. At x = 200: MC = 40 + 20 = 60.
- Exact cost of the 201st unit: C(201) − C(200) = 40 + 0.05(201² − 200²) = 40 + 20.05 = 60.05, so MC is a good estimate.
- MR = R′(x) = 150 − 0.4x. At x = 200: MR = 150 − 80 = 70.
- Marginal profit = MR − MC = 70 − 60 = 10 > 0.
MC = Rs. 60, MR = Rs. 70 at 200 units: one more unit adds about Rs. 10 profit, so output should rise.
Price elasticity of demand
Demand for bottled tea is q = 200 − 4p. Find the price elasticity at p = 30 and say whether a price rise would raise revenue.
- At p = 30: q = 200 − 4(30) = 80.
- dq/dp = −4.
- E = (p/q) × dq/dp = (30/80) × (−4) = −1.5.
- |E| = 1.5 > 1, so demand is elastic: quantity falls by a bigger percentage than price rises.
E = −1.5 (elastic): a price rise would reduce revenue.
When a Sri Lankan garment exporter asks 'what will one more order of shirts add to cost?', it wants marginal cost, the derivative of the cost function. Comparing it with marginal revenue says whether to expand.
Short answer: “Differentiate y = 3x⁴ − 2x + 7.” dy/dx = 12x³ − 2.
Short answer: “Define marginal cost.” The rate of change of total cost with output, MC = dC/dx: roughly the cost of one more unit.
Essay: “C = 2,000 + 40x + 0.05x², R = 150x − 0.2x². Find MC, MR and marginal profit at 200 units and interpret.” MC = 60, MR = 70, marginal profit 10: raise output.
Power rule: bring the power down, then subtract one. Constants (like fixed cost) disappear. Marginal = derivative of the total.