SQA · Advanced Higher Mathematics · 2026 diet
Advanced Higher (AH) Maths Formula Sheet
Every formula the SQA prints at the front of the AH 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 (AH) Maths exam. It runs to two pages at the front of the question paper. On it: nine standard derivatives, five standard integrals, the arithmetic and geometric series sums, the sums of r, r² and r³, the binomial theorem, the Maclaurin expansion, de Moivre's theorem, the vector product and the anticlockwise rotation matrix.
That is the whole list. The product rule is not on it. Neither is the quotient rule, integration by parts, partial fractions, the nth term of a sequence, the sum to infinity, 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 (© Scottish Qualifications Authority, reproduced here as a study reference). It is reprinted inside every past paper, so you can check it against any recent one on our Advanced Higher Maths past papers page, or open the 2024 Paper 1 (PDF). Last checked on 25 September 2026 against the 2022 and 2024 question papers.
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 |
| ln x | 1 ⁄ x |
| ex | ex |
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 |
|---|---|
| Arithmetic series | Sn = (n⁄2)[2a + (n − 1)d] |
| Geometric series | Sn = a(1 − rn) ⁄ (1 − r), r ≠ 1 |
| Σr=1n r | n(n + 1) ⁄ 2 |
| Σr=1n r² | n(n + 1)(2n + 1) ⁄ 6 |
| Σr=1n r³ | n²(n + 1)² ⁄ 4 |
Both series sums are printed, but the nth terms and the sum to infinity are not (see below). The sums of r, r² and r³ 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) |
Matrix transformation
| Anticlockwise rotation through θ about the origin | [[cos θ, −sin θ], [sin θ, cos θ]] |
Reflections and dilations are not printed. Learn those (see the matrices box below).
What is not on the sheet
None of this is printed in the exam. You need it from memory. Formulas you can derive from the sheet (like the standard Maclaurin series) are marked as such.
Differentiation
- Product rule: (uv)′ = u′v + uv′
- Quotient rule: (u⁄v)′ = (u′v − uv′) ⁄ v²
- Chain rule: d⁄dx [f(g(x))] = f′(g(x)) × g′(x)
- Implicit differentiation: d⁄dx (yn) = nyn−1 dy⁄dx, and d⁄dx (xy) = y + x dy⁄dx
- Parametric: dy⁄dx = (dy⁄dt) ⁄ (dx⁄dt)
- From Higher: xn → nxn−1, sin x → cos x, cos x → −sin x
Integration
- By parts: ∫ u (dv⁄dx) dx = uv − ∫ v (du⁄dx) dx
- Substitution: ∫ f(g(x)) g′(x) dx = ∫ f(u) du, where u = g(x)
- Partial fractions: A⁄(x − a) + B⁄(x − b); a repeated factor gives A⁄(x − a) + B⁄(x − a)²; an irreducible quadratic gives (Ax + B)⁄(x² + c). Divide first if the numerator has the higher or equal degree.
- Volume of revolution about the x-axis: V = π ∫ y² dx; about the y-axis: V = π ∫ x² dy
- From Higher: ∫ xn dx = xn+1⁄(n + 1) + C, ∫ sin ax dx = −(1⁄a) cos ax + C, ∫ cos ax dx = (1⁄a) sin ax + C
Sequences and series
- Arithmetic nth term: un = a + (n − 1)d
- Geometric nth term: un = arn−1
- Sum to infinity: S∞ = a ⁄ (1 − r), |r| < 1
- The two finite sums, Sn for arithmetic and geometric series, are printed. See Summations above.
Standard Maclaurin series
- Derivable from the printed expansion, but faster to recall:
- ex = 1 + x + x²⁄2! + x³⁄3! + …
- sin x = x − x³⁄3! + x5⁄5! − …
- cos x = 1 − x²⁄2! + x4⁄4! − …
- ln(1 + x) = x − x²⁄2 + x³⁄3 − …, valid for −1 < x ≤ 1
Complex numbers
- Modulus: |z| = √(x² + y²)
- Argument: tan θ = y⁄x, choosing the quadrant from the signs of x and y
- Polar form: z = r(cos θ + i sin θ)
- Product: z1z2 = r1r2[cos(θ1 + θ2) + i sin(θ1 + θ2)]
- Quotient: z1⁄z2 = (r1⁄r2)[cos(θ1 − θ2) + i sin(θ1 − θ2)]
- Conjugate: z* = x − iy, and zz* = |z|². Divide by multiplying top and bottom by the conjugate of the denominator. Complex roots of a polynomial with real coefficients come in conjugate pairs.
- nth roots: z1⁄n = r1⁄n[cos((θ + 2πk)⁄n) + i sin((θ + 2πk)⁄n)], for k = 0, 1, …, n − 1
Matrices
- 2×2 determinant: det A = ad − bc
- 2×2 inverse: A−1 = (1⁄det A) [[d, −b], [−c, a]]
- 3×3 determinant, expanding along the first row: a(ei − fh) − b(di − fg) + c(dh − eg) for the matrix [[a, b, c], [d, e, f], [g, h, i]]
- Inverse of a product: (AB)−1 = B−1A−1
- Reflection in the x-axis: [[1, 0], [0, −1]]; in the y-axis: [[−1, 0], [0, 1]]
- Reflection in y = x: [[0, 1], [1, 0]]; in y = −x: [[0, −1], [−1, 0]]
- Dilation by scale factor k: [[k, 0], [0, k]]
- Gaussian elimination: row operations to reach upper triangular form, then back-substitute. A system is ill-conditioned when a small change in a coefficient causes a large change in the solution.
Vectors and 3D geometry
- Scalar product: a · b = |a||b| cos θ = a1b1 + a2b2 + a3b3
- Line, vector form: r = a + tb. Parametric: x = a1 + tb1, y = a2 + tb2, z = a3 + tb3. Symmetric: (x − a1)⁄b1 = (y − a2)⁄b2 = (z − a3)⁄b3
- Plane: ax + by + cz = d, with normal n = (a, b, c); through point a: n · (r − a) = 0
- Angle between two planes is the angle between their normals: cos θ = n1 · n2 ⁄ (|n1||n2|)
- Scalar triple product: a · (b × c), the 3×3 determinant of the components. Its modulus is the volume of the parallelepiped.
Differential equations
- First order, dy⁄dx + P(x)y = Q(x): integrating factor I(x) = e∫P(x)dx, then I(x)y = ∫ I(x)Q(x) dx
- Second order, ay″ + by′ + cy = 0: auxiliary equation am² + bm + c = 0
- Two real roots m1 ≠ m2: y = Aem1x + Bem2x
- One repeated root m: y = (A + Bx)emx
- Complex roots p ± qi: y = epx(A cos qx + B sin qx)
- With a right-hand side f(x): general solution = complementary function + particular integral. Try a polynomial of the same degree for a polynomial, Ceλx for eλx, and A sin kx + B cos kx for a sine or cosine. Multiply by x if the trial term is already in the complementary function.
Proof and number theory
- Induction: prove the statement for the first value (usually n = 1); assume it is true for n = k; show it is then true for n = k + 1; conclude it is true for all n from the first value on.
- Contradiction: assume the opposite of what you want to prove, then reach something impossible.
- Contrapositive: to prove “P implies Q”, prove “not Q implies not P”.
- Euclidean algorithm: write a = bq + r repeatedly, replacing (a, b) with (b, r). The last non-zero remainder is the gcd; work backwards to write it as ax + by.
- Number bases and divisibility: convert between bases by repeated division, and use a | b for “a divides b”.
Trigonometry carried over from Higher
- sin(A ± B) = sin A cos B ± cos A sin B
- cos(A ± B) = cos A cos B ∓ sin A sin B
- sin 2A = 2 sin A cos A
- cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A
- sin²A + cos²A = 1
The compound angle and double angle formulae are printed on the Higher list but not reprinted at Advanced Higher, so you are expected to know them. So is sin²A + cos²A = 1.
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.
- Do not memorise what is printed. The ln x and ex derivatives, both series sums and the rotation matrix are all on the list. Spend your revision time on the formulae that are not.
- Write the memorised ones down first. In the reading time, put the product rule, integration by parts, the nth terms and the sum to infinity 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?
Yes. A formulae list is printed at the front of the Advanced Higher Mathematics question paper and you can use it throughout both papers. You do not need to bring anything of your own, and you are not permitted to bring your own notes.
What is the AH Maths formula sheet?
AH is short for Advanced Higher. The “formula sheet” is the formulae list the SQA prints at the front of every Advanced Higher Mathematics question paper. It runs to two pages and you can refer to it throughout both papers.
What is on the Advanced Higher Maths formula sheet?
Nine standard derivatives (the three inverse trig functions, tan, cot, sec, cosec, ln x and ex), five standard integrals, the arithmetic and geometric series sums, the sums of r, r² and r³, the binomial theorem with the definition of nCr, the Maclaurin expansion, de Moivre's theorem, the vector product in both its geometric and determinant forms, and the matrix for an anticlockwise rotation about the origin. The full list is written out above.
Is integration by parts on the formula sheet?
No. Integration by parts is not printed, so you need to know it: ∫ u (dv⁄dx) dx = uv − ∫ v (du⁄dx) dx. The same goes for the product rule, the quotient rule and the method for partial fractions.
Are the arithmetic and geometric series formulas given?
Partly. The sums of the first n terms are printed: Sn = (n⁄2)[2a + (n − 1)d] for an arithmetic series and Sn = a(1 − rn) ⁄ (1 − r) for a geometric series. The nth terms (un = a + (n − 1)d and un = arn−1), the sum to infinity S∞ = a ⁄ (1 − r) and the condition |r| < 1 are not printed, so you need them from memory.
Is the rotation matrix on the AH Maths formula sheet?
Yes. The matrix for an anticlockwise rotation through θ about the origin, [[cos θ, −sin θ], [sin θ, cos θ]], is printed under “Matrix transformation”. Reflections and dilations are not, so learn those. For example, question 6 of 2024 Paper 1 asks for the matrix for a reflection in the x-axis.
Has the formula sheet changed for the 2026 exams?
The list printed in the 2022 and 2024 question papers is the same as the one on this page. SQA publishes the definitive version with each year's papers, so check the front pages of the most recent past paper before you sit the exam.
Where can I get the official SQA copy?
The formulae list is printed at the front of every Advanced Higher Maths question paper, so the simplest way to get an official copy is to open any recent paper on our Advanced Higher Maths past papers page. Course documents are also published on the SQA website.
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