Differential and Integral Calculus: formula sheet

Key formulas

Definition of limit

lim (x→a) f(x) = L, if LHL = RHL = L

LHL is the limit as x→a⁻ and RHL is the limit as x→a⁺. Both must be finite and equal.

Continuity at a point

lim (x→a) f(x) = f(a)

f(a) must be defined and the limit must exist.

Limit laws

lim [f(x) ± g(x)] = lim f(x) ± lim g(x); lim [f(x) × g(x)] = lim f(x) × lim g(x); lim [f(x) ÷ g(x)] = lim f(x) ÷ lim g(x)

Applies when each individual limit exists. For division, the limit of g(x) must not be 0.

Power standard limit

lim (x→a) (xⁿ − aⁿ) ÷ (x − a) = n·aⁿ⁻¹

Valid for any real n, with a > 0 when n is not an integer.

Exponential standard limit

lim (x→0) (eˣ − 1) ÷ x = 1

More generally, lim (x→0) (aˣ − 1) ÷ x = log_e a, for a > 0.

Logarithmic standard limit

lim (x→0) log_e(1 + x) ÷ x = 1

Natural log only.

Definition of e

lim (x→0) (1 + x)^(1/x) = e

Equivalent form: lim (n→∞) (1 + 1/n)ⁿ = e.

Limit at infinity

lim (x→∞) 1/xⁿ = 0, for n > 0

For a ratio of polynomials, divide the numerator and denominator by the highest power of x.

Derivative from first principles

f'(x) = lim (h→0) [f(x + h) − f(x)] ÷ h

Use only when the question says 'from first principles' or 'by definition'.

Derivative of a constant

d/dx (c) = 0

Applies to any number such as 5, e or log 2.

Power rule

d/dx (xⁿ) = n·xⁿ⁻¹

Works for any real n, including negative and fractional. So d/dx (1/x) = −1/x² and d/dx (√x) = 1/(2√x).

Constant multiple and sum rule

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

Differentiate each term separately.

Product rule

d/dx (u·v) = u·(dv/dx) + v·(du/dx)

Do not multiply the two derivatives.

Quotient rule

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

Valid where v ≠ 0. Numerator order matters: v·u' first, then minus u·v'.

Chain rule

dy/dx = (dy/du) × (du/dx), where y = f(u) and u = g(x)

Differentiate the outer function, keep the inner as it is, then multiply by the derivative of the inner.

Exponential derivatives

d/dx (eˣ) = eˣ; d/dx (aˣ) = aˣ·log a (a > 0)

d/dx (e^(f(x))) = e^(f(x))·f'(x) by the chain rule.

Log derivatives

d/dx (log x) = 1/x (x > 0); d/dx (log f(x)) = f'(x) ÷ f(x)

Here log is the natural logarithm. For base a, d/dx (logₐ x) = 1 ÷ (x log a).

Second derivative

d²y/dx² = d/dx (dy/dx) = f″(x)

Differentiate the first derivative once more with respect to x.

Third derivative

d³y/dx³ = d/dx (d²y/dx²) = f‴(x)

Keep going the same way for higher orders.

Power rule repeated

If y = xⁿ, then y′ = n xⁿ⁻¹ and y″ = n(n − 1) xⁿ⁻² (the y″ formula holds for any real n, and for a positive integer n it is meaningful for n ≥ 2). In general, the kth derivative is n(n − 1)(n − 2)...(n − k + 1) xⁿ⁻ᵏ.

For a positive integer n, the nth derivative of xⁿ is n! (a constant) and the (n + 1)th derivative is 0. For n = 1, y″ = 0, which the formula also gives.

Exponential

If y = eᵃˣ, then dⁿy/dxⁿ = aⁿ eᵃˣ

Each differentiation multiplies by a.

Logarithm

If y = log x, then y′ = 1/x, y″ = −1/x², y‴ = 2/x³

Valid for x > 0. Write 1/x as x⁻¹ and use the power rule.

Second derivative test

f′(x) = 0 and f″(x) < 0 → maximum; f′(x) = 0 and f″(x) > 0 → minimum

If f″(x) = 0 the test is inconclusive.

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.

Power rule

∫ xⁿ dx = xⁿ⁺¹ ÷ (n + 1) + C

Valid only for n ≠ −1.

Reciprocal rule

∫ (1/x) dx = log x + C (for x > 0; in general log |x| + C)

This is the case n = −1. Here log means natural logarithm (base e).

Exponential rule

∫ eˣ dx = eˣ + C

eˣ is unchanged by integration.

Exponential with constant base

∫ aˣ dx = aˣ ÷ log a + C (a > 0, a ≠ 1)

Log is natural log.

Constant

∫ k dx = kx + C

Special case of the power rule with n = 0.

Constant multiple rule

∫ k·f(x) dx = k ∫ f(x) dx

Take the constant outside.

Sum and difference rule

∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx

Integrate term by term. There is no such rule for products or quotients.

Linear inside function

∫ (ax + b)ⁿ dx = (ax + b)ⁿ⁺¹ ÷ [a(n + 1)] + C (n ≠ −1); ∫ eᵃˣ⁺ᵇ dx = eᵃˣ⁺ᵇ ÷ a + C; ∫ 1/(ax + b) dx = (1/a) log |ax + b| + C

Follows from substitution t = ax + b. Divide by a.

Check by differentiation

d/dx [∫ f(x) dx] = f(x)

Use it to verify an answer or to eliminate options.

Integration by parts

∫u·v dx = u·∫v dx − ∫( u′ · ∫v dx ) dx

u is the function chosen by ILATE. v is the other one. Do not forget the minus sign.

ILATE order

Inverse trig → Logarithmic → Algebraic → Trigonometric → Exponential

The function that appears earlier in this list is taken as u.

Integral of log x

∫log x dx = x·log x − x + C

Here log means natural log (base e). Take u = log x and v = 1.

Integral of x·eˣ

∫x·eˣ dx = eˣ(x − 1) + C

A standard result worth remembering for MCQs.

Special eˣ form

∫eˣ[f(x) + f′(x)] dx = eˣ·f(x) + C

Use it when the bracket is a function plus its own derivative.

Distinct linear factors

P(x)/((x−a)(x−b)) = A/(x−a) + B/(x−b)

Valid when the fraction is proper and a ≠ b. Find A and B by putting x = a and x = b.

Repeated linear factor

P(x)/((x−a)²(x−b)) = A/(x−a) + B/(x−a)² + C/(x−b)

A repeated factor needs one term for each power, up to the highest power.

Basic log integral

∫1/(x−a) dx = log|x−a| + C

Also ∫1/(x−a)² dx = −1/(x−a) + C.

Difference of squares form

∫1/(x² − a²) dx = (1/2a)·log|(x−a)/(x+a)| + C

Valid for a ≠ 0, on any interval where x ≠ ±a (the integrand is undefined at x = ±a). This is a ready result from partial fractions.

Fundamental theorem

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

Write the antiderivative in brackets with limits, then subtract lower from upper. No constant C is needed.

Power rule

∫ xⁿ dx = xⁿ⁺¹ ÷ (n + 1), for n ≠ −1

For n = −1, ∫ (1/x) dx = log x (natural log, x > 0).

Exponential rule

∫ eˣ dx = eˣ

Also ∫ eᵃˣ dx = eᵃˣ ÷ a for a ≠ 0.

Same limits

∫ from a to a of f(x) dx = 0

No width means no area.

Reversing limits

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

Swapping limits changes the sign.

Constant multiple and sum

∫ k·f(x) dx = k·∫ f(x) dx; ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx

Applies to the same limits.

Splitting the interval

∫ from a to b of f(x) dx = ∫ from a to c of f(x) dx + ∫ from c to b of f(x) dx

Useful when the curve changes sign or form at c.

Change of variable name

∫ from a to b of f(x) dx = ∫ from a to b of f(t) dt

The letter used does not matter.

King property

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

Used to simplify some integrals, such as when the result gives 2I = something simple.

Even and odd functions

∫ from −a to a of f(x) dx = 2 ∫ from 0 to a of f(x) dx if f is even; = 0 if f is odd

Even: f(−x) = f(x). Odd: f(−x) = −f(x).

Area under a curve

Area = ∫ from a to b of y dx, when y ≥ 0 on [a, b]

If y is below the axis, the integral is negative; take the absolute value for area.

Total cost from marginal cost

C(x) = ∫ MC dx + k, where k = fixed cost (since C(0) = k)

Extra cost from x = a to x = b is ∫ from a to b of MC dx.

Total revenue from marginal revenue

R(x) = ∫ MR dx, with R(0) = 0

No units sold means no revenue, so the constant is 0.

Consumer surplus

CS = ∫ from 0 to x₀ of D(x) dx − p₀·x₀

D(x) is the demand price at quantity x. (x₀, p₀) is the equilibrium point.

Producer surplus

PS = p₀·x₀ − ∫ from 0 to x₀ of S(x) dx

S(x) is the supply price at quantity x.

Quick revision

• d/dx (xⁿ) = n·xⁿ⁻¹ for any real n.
• d/dx (eˣ) = eˣ and d/dx (log x) = 1/x for x > 0 (log means natural log).
• d/dx (aˣ) = aˣ · log a for a > 0.
• Product rule: d(uv)/dx = u·dv/dx + v·du/dx.
• Quotient rule: d(u/v)/dx = (v·du/dx − u·dv/dx) ÷ v², for v ≠ 0.
• Chain rule: d/dx f(g(x)) = f'(g(x)) · g'(x).
• For maxima or minima, set f'(x) = 0. If f''(x) < 0 it is a maximum; if f''(x) > 0 it is a minimum.
• Marginal cost = d(Total Cost)/dx and marginal revenue = d(Total Revenue)/dx.
• ∫ xⁿ dx = xⁿ⁺¹ ÷ (n + 1) + C, for n ≠ −1; ∫ (1/x) dx = log x + C for x > 0.
• ∫ eˣ dx = eˣ + C, and ∫ aˣ dx = aˣ ÷ log a + C for a > 0, a ≠ 1.
• Integration by parts: ∫ u·v dx = u∫v dx − ∫ (du/dx · ∫v dx) dx.
• Definite integral: ∫ from a to b of f(x) dx = F(b) − F(a); the constant C cancels out.

Common mistakes

• Treating 0/0 as 0 or as 1. Fix: 0/0 is an indeterminate form. Simplify the expression first, then substitute again.
• Confusing the limit with the value of the function at the point. Fix: Remember that the limit describes nearby values. Continuity requires the limit to equal f(a).
• Writing d/dx (u·v) = u'·v' Fix: Always write u·v' + v·u'. Test with x·x: the rule gives 2x, while u'·v' gives 1.
• Reversing the numerator in the quotient rule Fix: Remember 'low d-high minus high d-low, over low squared'. The denominator's derivative term always comes second.
• Substituting the value of x after the first derivative and then differentiating again. Fix: Differentiating a number gives zero, which is wrong. Substitute only after the full derivative is found.
• Writing d²y/dx² as (dy/dx)². Fix: d²y/dx² means differentiate twice. (dy/dx)² means the first derivative squared. They are different.
• Writing MC as C(x) ÷ x Fix: Marginal means derivative. Average means divide by x.
• Keeping the fixed cost in the marginal cost Fix: Remove the constant term when differentiating. Keep it for AC.
• Forgetting the constant C in an indefinite integral. Fix: Always write + C at the end. In MCQs, the option with + C is the expected one.
• Using the power rule on 1/x, giving x⁰ ÷ 0. Fix: Check the power first. If it is −1, the answer is log x + C.

Exam tips

• Always try direct substitution first. Many MCQs are solved in under 20 seconds this way.
• Learn the n·aⁿ⁻¹ result and the three e-based standard limits. They are the most tested forms.
• For continuity questions on piecewise functions, equate LHL, RHL and f(a) and solve for the unknown constant.
• Check each option against your answer. Do not guess blindly, since a wrong answer loses 0.25 marks.
• Practise polynomial ratios at infinity until you can read the answer from the leading terms.
• Expect direct one-step questions: find dy/dx for a polynomial, a product, a quotient or e/log of an expression. Practise speed on these.
• Always simplify before substituting a value of x, and re-check the arithmetic, since options often contain sign traps.
• Learn the chain rule forms e^(f(x)) → e^(f(x))·f'(x) and log f(x) → f'(x)/f(x). They appear again in maxima, minima and marginal analysis.