Advanced Higher Maths Formula Sheet
Every formula the SQA prints at the front of the Advanced Higher Maths paper, written out below. Then the ones it leaves out, which you are expected to know from memory.
You get a formulae list in the Advanced Higher Maths exam. It runs to one page. On it: seven standard derivatives, five standard integrals, the sums of r, r² and r³, the binomial theorem, the Maclaurin expansion, de Moivre's theorem and the vector product.
That is the whole list. The product rule is not on it. Neither is the quotient rule, integration by parts, partial fractions, the arithmetic and geometric series formulas, or the determinant and inverse of a matrix. The second half of this page covers those.
The full SQA formulae list
This is the list as it appears at the front of the question paper. It is reprinted inside every past paper, so you can check it against any recent one on our Advanced Higher Maths past papers page.
Standard derivatives
| f(x) | f′(x) |
|---|---|
| sin−1x | 1 ⁄ √(1 − x²) |
| cos−1x | −1 ⁄ √(1 − x²) |
| tan−1x | 1 ⁄ (1 + x²) |
| tan x | sec² x |
| cot x | −cosec² x |
| sec x | sec x tan x |
| cosec x | −cosec x cot x |
Standard integrals
| f(x) | ∫ f(x) dx |
|---|---|
| 1 ⁄ √(a² − x²) | sin−1(x⁄a) + C |
| 1 ⁄ (a² + x²) | (1⁄a) tan−1(x⁄a) + C |
| sec²(ax) | (1⁄a) tan(ax) + C |
| 1 ⁄ x | ln |x| + C |
| eax | (1⁄a) eax + C |
Summations
| Sum | Result |
|---|---|
| Σr=1n r | n(n + 1) ⁄ 2 |
| Σr=1n r² | n(n + 1)(2n + 1) ⁄ 6 |
| Σr=1n r³ | n²(n + 1)² ⁄ 4 |
These three turn up in proof by induction and in series questions.
Binomial theorem
| Expansion | (a + b)n = Σr=0n nCr an−r br |
| Binomial coefficient | nCr = n! ⁄ (r!(n − r)!) |
Maclaurin expansion
| Series | f(x) = f(0) + f′(0)x + f″(0)x²⁄2! + f″′(0)x³⁄3! + f(4)(0)x4⁄4! + … |
The standard expansions of ex, sin x, cos x and ln(1 + x) are not printed. You work them out from this one.
De Moivre's theorem
| Theorem | [r(cos θ + i sin θ)]n = rn(cos nθ + i sin nθ) |
Vector product
| Definition | a × b = |a||b| sin θ n̂ |
| Determinant form | = i(a2b3 − a3b2) − j(a1b3 − a3b1) + k(a1b2 − a2b1) |
What is not on the sheet
None of this is printed in the exam. You need it from memory.
Differentiation
- Product rule: (uv)′ = u′v + uv′
- Quotient rule: (u⁄v)′ = (u′v − uv′) ⁄ v²
- Chain rule
- Implicit differentiation
- Parametric: dy⁄dx = (dy⁄dt) ⁄ (dx⁄dt)
Integration
- By parts: ∫ u (dv⁄dx) dx = uv − ∫ v (du⁄dx) dx
- Partial fractions: setting up and solving
- Volume of revolution: V = π ∫ y² dx
- Integration by substitution
Sequences and series
- Arithmetic nth term: un = a + (n − 1)d
- Arithmetic sum: Sn = (n⁄2)[2a + (n − 1)d]
- Geometric nth term: un = arn−1
- Geometric sum: Sn = a(1 − rn) ⁄ (1 − r)
- Sum to infinity: S∞ = a ⁄ (1 − r), |r| < 1
Complex numbers
- Modulus: |z| = √(x² + y²)
- Polar form and argument
- Conjugate and division by the conjugate
- nth roots of a complex number
Matrices
- 2×2 determinant: det A = ad − bc
- 2×2 inverse: A−1 = (1⁄det A) [[d, −b], [−c, a]]
- 3×3 determinant
- Gaussian elimination and ill-conditioning
Vectors and 3D geometry
- Scalar product: a · b = |a||b| cos θ
- Vector, parametric and symmetric equations of a line
- Equation of a plane, and the angle between planes
- Scalar triple product
Differential equations
- Integrating factor: I(x) = e∫P(x)dx
- Second-order: auxiliary equation and the three root cases
- Particular integral and complementary function
Proof and number theory
- The structure of proof by induction
- Proof by contradiction and by contrapositive
- Euclidean algorithm
- Number bases and divisibility
Compound angle and double angle identities carry over from Higher and are not reprinted at Advanced Higher either.
Using the sheet in the exam
- Read it before the exam. If the first time you see the list is in the hall, you will spend ten minutes looking for a formula that was never on it.
- Check the form each entry is given in. The integrals use a, not a number, so you have to match your question to the pattern first.
- Write the memorised ones down first. In the reading time, put the product rule, integration by parts and the two series formulas at the top of your booklet.
- Practise with it open. Sit past papers with the formulae page next to you, so you already know where on it to look.
Common questions
Do you get a formula sheet in the Advanced Higher Maths exam?
What is on the Advanced Higher Maths formula sheet?
Is integration by parts on the formula sheet?
Are the arithmetic and geometric series formulas given?
Has the formula sheet changed for the 2026 exams?
Where can I get the official SQA copy?
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