Calculus - Application in Business: formula sheet

Key formulas

Derivative of a constant

d/dx (c) = 0

Any fixed number, such as 7 or ₹500, has zero rate of change.

Power rule

d/dx (xⁿ) = n·xⁿ⁻¹

Works for any real n, including fractions and negatives. √x = x^(1/2) and 1/x = x⁻¹.

Constant multiple and sum rule

d/dx [a·f(x) ± b·g(x)] = a·f'(x) ± b·g'(x)

Differentiate each term separately.

Product rule

d/dx (u·v) = u·(dv/dx) + v·(du/dx)

Use when two functions of x are multiplied.

Quotient rule

d/dx (u ÷ v) = [v·(du/dx) − u·(dv/dx)] ÷ v²

For v ≠ 0. The numerator order matters: v·u' first, then minus u·v'.

Chain rule

dy/dx = (dy/du) × (du/dx)

For y = f(u) where u = g(x). Differentiate outside, then multiply by the derivative of the inside.

Exponential functions

d/dx (eˣ) = eˣ; d/dx (e^(f(x))) = e^(f(x)) · f'(x); d/dx (aˣ) = aˣ · log_e a

For a > 0. eˣ is its own derivative.

Logarithmic functions

d/dx (log_e x) = 1/x; d/dx (log_e f(x)) = f'(x) ÷ f(x)

For x > 0 and f(x) > 0. Here log means natural log (base e).

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.

Condition for a stationary point

f'(x) = 0

Solve this equation to get the critical points.

Test for maximum

f'(x) = 0 and f''(x) < 0

The function has a local maximum at that x.

Test for minimum

f'(x) = 0 and f''(x) > 0

The function has a local minimum at that x.

Profit

P(x) = R(x) − C(x)

Maximise this. At the optimum, marginal revenue equals marginal cost (R'(x) = C'(x)) with P''(x) < 0.

Average cost

AC = C(x) ÷ x

Differentiate AC itself to minimise it, not C(x).

Power rule

d/dx (xⁿ) = n·xⁿ⁻¹

The derivative of a constant is 0.

Point elasticity of demand

η = (p ÷ q) × (dq/dp)

Usually negative for a falling demand curve. Many questions ask for |η|, so say which you give. Use dq/dp, not dp/dq.

Elasticity from p as a function of q

η = (p ÷ q) × (1 ÷ (dp/dq))

Use this when demand is given as p = f(q). Take the reciprocal of dp/dq.

Classification

|η| > 1 elastic; |η| = 1 unitary; |η| < 1 inelastic

Revenue rises when price falls if demand is elastic. Revenue rises when price rises if demand is inelastic.

Revenue and marginal revenue

R = p × q; MR = dR/dq

Express R in terms of q alone before differentiating.

Profit and its condition

π = R − C; dπ/dq = 0 and d²π/dq² < 0 for maximum

dπ/dq = 0 is the same as MR = MC.

Marginal and average cost

MC = dC/dq; AC = C ÷ q

Where AC is at its minimum, MC = AC. Find it from d(AC)/dq = 0.

Cost minimization test

dC/dq = 0 and d²C/dq² > 0 for minimum

Apply to total cost or to average cost, whichever the question names.

Tax revenue

T = t × q, where t is the tax per unit

Adding a per-unit tax raises total cost by t × q. Check whether the question maximizes profit or tax revenue.

Economic order quantity

TC = (D ÷ Q) × S + (Q ÷ 2) × H; minimum at Q = √(2DS ÷ H)

D is annual demand, S is ordering cost per order, H is holding cost per unit per year.

Power rule

∫ xⁿ dx = xⁿ⁺¹ ÷ (n + 1) + C

Valid for n ≠ -1. Raise the power by 1, then divide by the new power.

Constant

∫ k dx = kx + C

For example, ∫ 5 dx = 5x + C.

Reciprocal

∫ (1/x) dx = log x + C

This is the natural log, for x > 0. It is the case n = -1 that the power rule cannot handle.

Exponential

∫ eˣ dx = eˣ + C and ∫ e^(ax) dx = e^(ax) ÷ a + C

Divide by the coefficient of x when it is not 1.

Sum and constant multiple

∫ [a·f(x) ± b·g(x)] dx = a∫f(x) dx ± b∫g(x) dx

Integrate term by term. Constants come out in front.

Definite integral

∫ from a to b of f(x) dx = F(b) − F(a)

F is any antiderivative of f. The constant C cancels, so ignore it.

Total cost from marginal cost

C(x) = ∫ MC dx, with C(0) = fixed cost

Use the fixed cost to find the constant.

Total revenue from marginal revenue

R(x) = ∫ MR dx, with R(0) = 0

The constant is zero. Demand price is then p = R(x) ÷ x.

Consumer surplus

CS = ∫ from 0 to x₀ of D(x) dx − p₀x₀

D(x) is the demand curve. (x₀, p₀) is the equilibrium or given point.

Producer surplus

PS = p₀x₀ − ∫ from 0 to x₀ of S(x) dx

S(x) is the supply curve. Note the order is the reverse of CS.

Quick revision

• Power rule: d(xⁿ)/dx = n·xⁿ⁻¹; the derivative of a constant is 0.
• Product rule: d(uv)/dx = u·v′ + v·u′.
• Quotient rule: d(u/v)/dx = (v·u′ − u·v′) ÷ v².
• Marginal cost is the derivative of total cost; marginal revenue is the derivative of total revenue.
• Profit = revenue − cost. Maximum profit needs MR = MC (first-order condition) and the second derivative of profit negative (second-order condition).
• Stationary point: first derivative = 0.
• Second derivative negative means maximum; positive means minimum.
• At minimum average cost, AC = MC. Find the point from d(AC)/dx = 0 and confirm with the second derivative.
• Price elasticity of demand = −(dq/dp) × (p/q); compare its size with 1.
• ∫xⁿ dx = xⁿ⁺¹ ÷ (n+1) + C, for n ≠ −1.
• Integrating marginal cost gives total cost; fixed cost fixes the constant.
• Always check units and whether the question asks for output, price or profit value.

Common mistakes

• Writing the derivative of eˣ as x·eˣ⁻¹. Fix: The power rule is only for xⁿ with a fixed power. For eˣ the derivative is eˣ.
• Differentiating a product as u'·v'. Fix: Use u·v' + v·u'. Test with x·x: the answer must be 2x, not 1.
• Treating average cost and marginal cost as the same. 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. Fix: Multiply first to get R(x) = 100x − 2x², then differentiate to get MR = 100 − 4x.
• Stopping after f'(x) = 0 and calling the answer the maximum value. Fix: Always put x back into f(x) if the question asks for the maximum or minimum value.
• Skipping the second derivative test and guessing max or min. Fix: Check the sign of f''(x) at the critical point, or use the sign of a for a quadratic.
• Using dp/dq instead of dq/dp in the elasticity formula. Fix: Elasticity needs dq/dp. If you have dp/dq, take its reciprocal first, then multiply by p ÷ q.
• Leaving out the sign or reporting a negative value as 'elasticity less than 1'. Fix: Compute η, then compare |η| with 1. State that demand is elastic, unitary or inelastic according to |η|.
• Forgetting the constant of integration in an indefinite integral. Fix: Write + C immediately. In cost problems, replace C with the fixed cost. In revenue problems, C = 0.
• Using the power rule on 1/x, giving x⁰ ÷ 0. Fix: Remember ∫ (1/x) dx = log x. Check the power before applying the rule.

Exam tips

• Questions are usually one-step or two-step applications. Identify the rule within five seconds and move on.
• Keep the options in view. Many wrong options come from a missed chain factor or a reversed quotient rule.
• Substitute the given x only after differentiating. Substituting first turns the function into a constant with derivative zero.
• Convert roots and fractions to powers before differentiating, for example 1/√x = x^(−1/2).
• With no negative marking, always mark an answer. If stuck, eliminate options using a simple test value.
• 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.