What does "marginal" mean?
The word marginal means "at the edge" or "one more". Marginal cost is the extra money you spend to make one more item.
If making 4 items costs 16 and making 5 items costs 25, then the marginal cost of the 5th item is 25 − 16 = 9.
We write it as: marginal cost = change in cost ÷ change in quantity = ΔC ÷ Δq.
From "one more" to "a tiny bit more"
Sometimes we cannot make exactly one more. A factory may add a tiny bit of milk or 0.1 of a litre. So we use a small step h.
Extra cost per item = (C(q + h) − C(q)) ÷ h.
This is the slope of the line through two points on the cost curve. Now make h smaller: 1, then 0.5, then 0.1, then 0.01. The slope stops changing much. The number it gets close to is the derivative of C at q, written C′(q) or dC/dq.
Finding the derivative with a rule
You do not need to shrink h every time. There are easy rules:
- For C = q², the derivative is 2q.
- For C = a·q (a straight line), the derivative is the number a.
- A fixed cost (like rent, a plain number) has derivative 0. It does not change when you make one more.
- For C = q³, the derivative is 3q².
Pattern: for qⁿ, bring n down in front and make the power one smaller: n·qⁿ⁻¹.
Using it in the economy
Marginal cost (MC) is the derivative of total cost. Marginal revenue (MR) is the derivative of total revenue, the money you get from one more sale.
A business should make one more item as long as MR is bigger than MC. It stops when MR = MC. That is the idea behind finding the best quantity.
Try it
In the 3D, set q = 4 and h = 1. Read the slope. Now slide h down to 0.1. Read the slope again. Write both numbers. Then compare with 2q = 8. Predict first: what will the slope be at q = 6? Check it.
At home: a pack of 10 pens costs ₹50 and 11 pens cost ₹54. What did the 11th pen cost you?
Key formulas and definitions
- Marginal cost = ΔC ÷ Δq
- Slope with a step h = (C(q + h) − C(q)) ÷ h
- Derivative: C′(q) = value the slope gets close to as h → 0
- C = q² → C′ = 2q; C = aq → C′ = a; fixed cost → 0
- qⁿ → n·qⁿ⁻¹
Worked examples
1. Making 4 items costs ₹16 hundred and 5 items cost ₹25 hundred. Find the marginal cost of the 5th item.
Marginal cost = 25 − 16 = 9 (₹9 hundred).
2. C = q². Find the extra cost of going from q = 6 to q = 7.
C(6) = 36, C(7) = 49. Extra = 49 − 36 = 13.
3. C = q². Find the slope with step h = 0.5 at q = 4.
C(4.5) = 20.25, C(4) = 16. Slope = (20.25 − 16) ÷ 0.5 = 8.5.
4. C = q². Find C′(4) with the rule.
C′ = 2q, so C′(4) = 2 × 4 = 8. The step slopes 9, 8.5, 8.1… get close to 8.
5. Total cost is C = 3q + 20. What is the marginal cost? Why?
C′ = 3. The ₹20 is a fixed cost and has derivative 0. Every extra item adds ₹3.
6. Revenue is R = 10q and cost is C = q². Find q where MR = MC.
MR = 10 and MC = 2q. Set 10 = 2q, so q = 5. Making more than 5 costs more than it earns.
Common mistakes
- Thinking marginal cost is the total cost. It is only the extra cost of one more.
- Using C(q + 1) divided by C(q). We subtract, not divide.
- Forgetting to divide by h when the step is not 1.
- Treating a fixed cost as if it grows with q. Its derivative is 0.