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Fundamentals of Business Mathematics and Statistics · Calculus - Application in Business

Cost, Revenue and Profit Functions in Business Calculus

Updated 10 October 2026 · Fact-checked

A cost function C(x) gives total cost for x units, a revenue function R(x) gives total sales income, and profit P(x) = R(x) − C(x). Marginal cost and marginal revenue are their derivatives, C′(x) and R′(x). Average cost is C(x) ÷ x. Form the function first, then differentiate.

Understand Cost, Revenue and Profit Functions

A function links the number of units produced or sold, x, to a money value. In business calculus you meet three: cost, revenue and profit.

The total cost function C(x) usually has two parts: a fixed cost that does not change with output (rent, salaries) and a variable cost that grows with x. For example, C(x) = 5,000 + 40x means ₹5,000 fixed and ₹40 per unit variable.

The revenue function R(x) is price times quantity. If the price per unit is p, then R(x) = p × x. When the price depends on demand, such as p = 100 − 2x, you must put that price into R first: R(x) = (100 − 2x)x = 100x − 2x².

Profit is revenue minus cost: P(x) = R(x) − C(x). A positive value is profit and a negative value is loss. Break-even happens where P(x) = 0, that is R(x) = C(x).

Marginal means the rate of change at a given output. It is found by differentiation. Marginal cost C′(x) is roughly the extra cost of producing one more unit. Marginal revenue R′(x) is roughly the extra revenue from selling one more unit. Average cost is different: it is total cost divided by units, C(x) ÷ x.

Key formulas to remember

Total cost
C(x) = Fixed cost + Variable cost
Fixed cost is the constant term in C(x).
Revenue
R(x) = p × x
If p depends on x (demand function), substitute it before differentiating.
Profit
P(x) = R(x) − C(x)
Break-even where P(x) = 0.
Average cost
AC = C(x) ÷ x
Divide the whole cost function by x, including the fixed part.
Average revenue
AR = R(x) ÷ x = p
Average revenue equals price per unit.
Marginal cost
MC = dC/dx = C′(x)
Fixed cost disappears on differentiation.
Marginal revenue
MR = dR/dx = R′(x)
Differentiate R(x), not the price p.
Marginal profit
P′(x) = MR − MC
Profit is maximum where MR = MC, provided P″(x) < 0.
Power rule
d/dx (xⁿ) = n·xⁿ⁻¹
The derivative of a constant is 0.

How to solve Cost, Revenue and Profit Functions questions

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

  1. 1Read what is given: a cost function, a demand or price function, or fixed and variable costs. Note the value of x asked for.
  2. 2Write C(x). Add fixed cost to variable cost per unit times x if it is not given directly.
  3. 3Write R(x) = price × x. If price is a function of x, substitute it and expand.
  4. 4Form P(x) = R(x) − C(x) if profit is needed. Open the brackets carefully and change every sign of C(x).
  5. 5Differentiate the required function term by term using the power rule to get MC, MR or marginal profit.
  6. 6For average values, divide the original function by x before any differentiation.
  7. 7Put in the given value of x. Check the units and that you answered what was asked (MC, AC, MR or profit).

Quickest way: Differentiate first, then substitute

When to use it: Use when the question asks for a marginal value at a specific output and options are numeric.

  1. Get the function in the form ax² + bx + c.
  2. Differentiate mentally: 2ax + b.
  3. Put in x and compute one number.
  4. For average cost, compute C(x) at the given x and divide by x directly, with no differentiation.
  5. Match the result to the options and drop any option that equals the average when marginal was asked.

Common mistakes in Cost, Revenue and Profit Functions

  • Treating average cost and marginal cost as the same.

    Both are per-unit ideas, so they look alike.

    Fix: Average cost = C(x) ÷ x. Marginal cost = C′(x). Calculate the one the question names.

  • Differentiating R = p × x with p treated as a constant when p = 100 − 2x.

    Students skip the step of forming R(x) from the demand function.

    Fix: Multiply first to get R(x) = 100x − 2x², then differentiate to get MR = 100 − 4x.

  • Keeping the fixed cost in marginal cost.

    Students forget that the derivative of a constant is zero.

    Fix: Drop the constant term when finding MC. It stays only in total and average cost.

  • Sign errors in P(x) = R(x) − C(x).

    The minus sign is not applied to every term of C(x).

    Fix: Put C(x) in brackets and change each sign before combining like terms.

  • Dividing only the variable part by x for average cost.

    Students think fixed cost is not per-unit.

    Fix: Divide the entire C(x) by x, so a fixed cost of 5,000 becomes 5,000 ÷ x.

  • Substituting x before differentiating.

    Wanting to simplify early.

    Fix: Differentiate the function first, then substitute the value of x.

Worked examples

Example 1

The cost function of a firm is C(x) = 2x² + 30x + 4,000 (in ₹). Find the marginal cost at x = 25 and the average cost at x = 25.

Show the solution
  1. MC = C′(x) = 4x + 30.
  2. At x = 25, MC = 4 × 25 + 30 = 130.
  3. C(25) = 2 × 625 + 30 × 25 + 4,000 = 1,250 + 750 + 4,000 = 6,000.
  4. AC = 6,000 ÷ 25 = 240.

Answer: Marginal cost = ₹130 and average cost = ₹240.

Example 2

A firm's demand function is p = 80 − 2x and its cost function is C(x) = 10x + 200. Find the profit function, the marginal revenue at x = 10, and the output at which marginal profit is zero.

Show the solution
  1. R(x) = p × x = (80 − 2x)x = 80x − 2x².
  2. P(x) = R(x) − C(x) = 80x − 2x² − 10x − 200 = −2x² + 70x − 200.
  3. MR = R′(x) = 80 − 4x. At x = 10, MR = 80 − 40 = 40.
  4. P′(x) = −4x + 70. Setting P′(x) = 0 gives 4x = 70, so x = 17.5.
  5. Check: P″(x) = −4, which is negative, so this point gives maximum profit.

Answer: P(x) = −2x² + 70x − 200; MR at x = 10 is ₹40; marginal profit is zero at x = 17.5 units.

Exam tips

  • Read the last line first to see whether the question wants total, average or marginal value. Examiners often put the wrong one among the options.
  • When a demand function is given, always form R(x) = p × x before doing anything else.
  • If the cost is given as fixed plus per-unit cost, MC is simply the per-unit cost when C is linear.
  • Do the arithmetic once, carefully. There is no negative marking, so always attempt every question, but a quick sign check saves marks.
  • Remember MR = MC as the profit-maximising condition, since maxima questions often build on these functions.

Practice questions from Calculus - Application in Business

Cost, Revenue and Profit Functions in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Cost, Revenue and Profit Functions: frequently asked questions

What is the difference between average cost and marginal cost?

Average cost is total cost divided by the number of units, C(x) ÷ x. Marginal cost is the derivative C′(x), the rate at which total cost changes with output. Average cost includes fixed cost, while marginal cost does not.

How do I find marginal revenue from a demand function?

Multiply the demand function p by x to get R(x). Then differentiate R(x) with respect to x. For p = 100 − 2x, R = 100x − 2x² and MR = 100 − 4x.

How do I find the profit function from cost and revenue?

Subtract: P(x) = R(x) − C(x). Place C(x) in brackets so every term changes sign, then combine like terms. The result is usually a quadratic in x.

Does marginal cost equal the cost of the next unit exactly?

It is an approximation. Marginal cost C′(x) is the instantaneous rate of change, which is close to the actual cost of producing one more unit. In exam questions, you use the derivative value as marginal cost.