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Quantitative Aptitude · Differential and Integral Calculus

Applications of Differentiation: Cost, Revenue and Marginal Analysis

Updated 1 October 2026 · Fact-checked

Marginal cost is the derivative of the total cost function, C'(x). Marginal revenue is the derivative of the revenue function, R'(x). Average cost is C(x) ÷ x. Profit is maximised where MR = MC, provided the second derivative of profit is negative. Differentiate, set equal, solve for x, then check.

Understand Applications of Differentiation: Cost, Revenue and Marginal Analysis

A business has a total cost function C(x), where x is the number of units produced. It tells you the full cost of making x units. A revenue function R(x) tells you the money earned from selling x units.

Marginal cost (MC) is the extra cost of producing one more unit. In calculus, it is the rate of change of total cost, so MC = dC/dx. In the same way, marginal revenue (MR) = dR/dx. At Foundation level, you treat these derivatives as the marginal values directly.

Average cost (AC) is cost per unit: AC = C(x) ÷ x. It is not a derivative. Marginal cost is the cost of the next unit, while average cost is the cost spread over all units. Students lose marks by mixing these two.

Profit P(x) = R(x) − C(x). To find the output that maximises profit, differentiate P, set P'(x) = 0 (which means MR = MC), and solve. Then confirm it is a maximum using P''(x) < 0. If revenue comes from a price-demand relation p = f(x), then R = p × x, so build R first and then differentiate.

Key formulas to remember

Marginal cost
MC = dC/dx = C'(x)
Derivative of the total cost function. Fixed cost disappears on differentiation.
Marginal revenue
MR = dR/dx = R'(x)
Derivative of the total revenue function.
Average cost
AC = C(x) ÷ x
Includes fixed cost. Not a derivative.
Revenue from price
R(x) = p × x
If price depends on x, substitute p in terms of x first.
Profit
P(x) = R(x) − C(x)
Marginal profit is P'(x) = MR − MC.
Profit maximisation
P'(x) = 0, i.e. MR = MC, and P''(x) < 0
The second condition confirms a maximum.
Power rule
d/dx (axⁿ) = a·n·xⁿ⁻¹
The constant term differentiates to 0.

How to solve Applications of Differentiation: Cost, Revenue and Marginal Analysis questions

Use this order for any cost, revenue or profit question.

  1. 1Read what is asked: total cost, MC, MR, AC, profit, or the output for maximum profit.
  2. 2Write the needed function. If price is given as p = f(x), form R = p·x. If profit is needed, form P = R − C.
  3. 3Differentiate using the power rule. Drop constants.
  4. 4If a specific output is given, substitute that x into the derivative. For AC, substitute into C(x) and divide by x.
  5. 5For maximum profit, set P'(x) = 0 (or MR = MC) and solve for x.
  6. 6Check P''(x) < 0. If the equation gives two roots, reject the one that fails or is negative.
  7. 7Put x back into the right function if the question asks for the maximum profit, not the output.
  8. 8Match your value with the four options.

Quickest way: Differentiate, substitute, eliminate

When to use it: Use in MCQs where C(x) or R(x) is given and the options are distinct numbers.

  1. For MC at a given x, differentiate mentally term by term and substitute. Do not expand anything.
  2. For AC, divide C(x) by x term by term first, then substitute.
  3. For optimum output, solve MR = MC directly. Skip finding P unless asked.
  4. For a quadratic profit with negative x² coefficient, the maximum is at x = −b ÷ 2a, with no calculus needed.
  5. Test options by substitution if the equation is hard. Eliminate negative outputs.
  6. If you are unsure after one minute, skip. A wrong answer costs 0.25 marks.

Common mistakes in Applications of Differentiation: Cost, Revenue and Marginal Analysis

  • Writing MC as C(x) ÷ x

    Students confuse marginal cost with average cost.

    Fix: Marginal means derivative. Average means divide by x.

  • Keeping the fixed cost in the marginal cost

    Forgetting that the derivative of a constant is zero.

    Fix: Remove the constant term when differentiating. Keep it for AC.

  • Differentiating R = p·x without first substituting p

    Students treat p as a constant when p depends on x.

    Fix: Write R purely in x, then differentiate.

  • Not checking the second derivative

    Stopping once P'(x) = 0 gives an answer.

    Fix: Find P''(x). A negative value means maximum. Positive means minimum.

  • Giving output when profit was asked

    Students stop at x after solving MR = MC.

    Fix: Reread the question and substitute x into P(x) if profit is required.

  • Substituting x into C'(x) when AC is asked

    Rushing and mixing up the formulas.

    Fix: Underline the term asked before starting.

Worked examples

Example 1

The total cost of producing x units is C(x) = 2x³ − 5x² + 40x + 500. The marginal cost at x = 5 is: (a) ₹65 (b) ₹75 (c) ₹90 (d) ₹140

Show the solution
  1. MC = C'(x) = 6x² − 10x + 40.
  2. At x = 5: 6 × 25 = 150.
  3. 10 × 5 = 50.
  4. MC = 150 − 50 + 40 = 140.

Answer: (d) ₹140

Example 2

For C(x) = 0.1x² + 20x + 1,000, the average cost at x = 50 units is: (a) ₹20 (b) ₹25 (c) ₹45 (d) ₹55

Show the solution
  1. C(50) = 0.1 × 2,500 + 20 × 50 + 1,000.
  2. = 250 + 1,000 + 1,000 = 2,250.
  3. AC = 2,250 ÷ 50 = 45.

Answer: (c) ₹45

Example 3

Revenue is R(x) = 80x − x² and cost is C(x) = 20x + 100. The output that maximises profit is: (a) 20 (b) 25 (c) 30 (d) 40

Show the solution
  1. P(x) = 80x − x² − 20x − 100 = 60x − x² − 100.
  2. P'(x) = 60 − 2x.
  3. Set P'(x) = 0, so x = 30.
  4. P''(x) = −2, which is negative, so this is a maximum.
  5. Check with MR = MC: 80 − 2x = 20 gives x = 30.

Answer: (c) 30

Exam tips

  • Read the last line first. Many MCQs ask for MC, AC or profit and the options include the wrong ones as traps.
  • If the cost function has a constant, expect a question that tests whether you drop it for MC.
  • For profit questions with quadratic functions, use MR = MC and solve a linear equation. It is faster.
  • Plug the options back in only when solving gets long. Check the sign of P'' mentally.
  • Keep units straight: output is in units, cost and revenue in rupees.

Practice questions from Differential and Integral Calculus

Applications of Differentiation: Cost, Revenue and Marginal Analysis: frequently asked questions

How do I find marginal cost from a total cost function?

Differentiate the total cost function with respect to x. Constants such as fixed cost vanish. Then substitute the required output into the derivative.

What is the difference between average cost and marginal cost?

Average cost is total cost divided by units produced. Marginal cost is the derivative of total cost, the cost of one more unit. They generally have different values at any output.

When is profit maximum?

Profit is maximum where MR = MC, i.e. P'(x) = 0, and P''(x) < 0. The second condition makes sure it is a maximum, not a minimum.

Is fixed cost included in marginal cost?

No. Fixed cost is a constant, and its derivative is zero. It does count in total cost and average cost.