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Advanced Higher Maths formula sheet: what the SQA formulae list gives you

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Formula sheet

The card below opens today's Advanced Higher Maths paper with this list in the drawer. The contents follow in our own layout. The list was the same in the 2025 and 2026 papers. Check it against the list in a recent paper, such as the 2026 Paper 1, before you rely on this page.

Formula sheet

Advanced Higher Maths

Open formula sheet

Standard derivatives

\frac{d}{dx}(\sin^{-1}x)=\frac{1}{\sqrt{1-x^2}}derivative of inverse sine
\frac{d}{dx}(\cos^{-1}x)=-\frac{1}{\sqrt{1-x^2}}derivative of inverse cosine
\frac{d}{dx}(\tan^{-1}x)=\frac{1}{1+x^2}derivative of inverse tangent
\frac{d}{dx}(\tan x)=\sec^2xderivative of tan x
\frac{d}{dx}(\cot x)=-\csc^2xderivative of cot x
\frac{d}{dx}(\sec x)=\sec x\tan xderivative of sec x
\frac{d}{dx}(\csc x)=-\csc x\cot xderivative of cosec x
\frac{d}{dx}(\ln x)=\frac{1}{x}derivative of ln x
\frac{d}{dx}(e^x)=e^xderivative of e^x

Standard integrals

\int\sec^2(ax)\ dx=\frac{1}{a}\tan(ax)+cintegral of sec² ax
\int\frac{1}{\sqrt{a^2-x^2}}\ dx=\sin^{-1}\left(\frac{x}{a}\right)+cintegral giving inverse sine
\int\frac{1}{a^2+x^2}\ dx=\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right)+cintegral giving inverse tangent
\int\frac{1}{x}\ dx=\ln|x|+cintegral of 1/x
\int e^{ax}\ dx=\frac{1}{a}e^{ax}+cintegral of e^(ax)

Summations

S_n=\frac{n}{2}\left[2a+(n-1)d\right]sum of an arithmetic series
S_n=\frac{a(1-r^n)}{1-r},\ r\ne1sum of a geometric series
\sum_{r=1}^{n}r=\frac{1}{2}n(n+1)sum of the first n integers
\sum_{r=1}^{n}r^2=\frac{1}{6}n(n+1)(2n+1)sum of the first n squares
\sum_{r=1}^{n}r^3=\frac{1}{4}n^2(n+1)^2sum of the first n cubes

Binomial theorem

(a+b)^n=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^rexpansion of (a + b)ⁿ
\binom{n}{r}=\frac{n!}{r!(n-r)!}the binomial coefficient

Maclaurin expansion

f(x)=f(0)+xf^{\prime}(0)+\frac{x^2}{2!}f^{\prime\prime}(0)+\frac{x^3}{3!}f^{\prime\prime\prime}(0)+\cdotsMaclaurin series of f(x)

De Moivre's theorem

\left[r(\cos\theta+i\sin\theta)\right]^n=r^n(\cos n\theta+i\sin n\theta)de Moivre's theorem

Vector product

a\times b=|a||b|\sin\theta\ \hat{n}vector product from the angle θ, with n a unit vector perpendicular to both
a\times b=(a_2b_3-a_3b_2)i-(a_1b_3-a_3b_1)j+(a_1b_2-a_2b_1)kvector product from components

Matrix transformation

An anti-clockwise rotation through an angle θ about the origin has this matrix:

Rotation matrixColumn 1Column 2
Row 1cos θ−sin θ
Row 2sin θcos θ

What is not on the Advanced Higher Maths formula sheet

  • The Higher formulae. The circle equations, the scalar product, and the addition and double angle formulae are not printed again. They are still examined.
  • Differentiation rules. The product, quotient and chain rules, implicit and parametric differentiation, and logarithmic differentiation.
  • Integration methods. Integration by parts, substitution and partial fractions: the table gives results, not methods.
  • Series. The nth term of a sequence, the sum to infinity of a geometric series, and the conditions for convergence.
  • Complex numbers. Modulus, argument, conjugates and how to find roots of a complex number.
  • Matrices. Determinants, inverses and the method of row operations. Only the rotation matrix is given.
  • Differential equations. The integrating factor and the methods for first and second order equations.
  • Proof. Induction and the structure of a proof.

How the papers use the list

The course specification gives 35 marks and 1 hour for Paper 1, where you may not use a calculator, and 80 marks and 2 hours 30 minutes for Paper 2, where you may. The list is printed in both. In the 2027 timetable both are on Monday 17 May: Paper 1 from 12:30 to 13:30 and Paper 2 from 14:00 to 16:30 (Exam Timetable 2027, version 02).

PaperMarksTimeCalculator
Paper 1351 hourNo
Paper 2802 hours 30 minutesYes

Our Advanced Higher Maths guide covers the course and the grades.

Using the formulae list under exam pressure

  1. Name the method before you look at the list. The table tells you the result, not the method.
  2. Copy the result you need into your working.
  3. Substitute carefully, particularly for a in the integrals and the chain rule.
  4. Leave the answer exact unless the question asks for a decimal.

Differentiating an inverse tangent

Differentiate tan⁻¹(3x). The list gives the derivative of tan⁻¹ x as 1/(1 + x²). The chain rule multiplies by the derivative of 3x, which is 3, and replaces x with 3x:

\frac{d}{dx}\left(\tan^{-1}3x\right)=\frac{3}{1+(3x)^2}=\frac{3}{1+9x^2}
practice
Pick an answer, then check it.
What is the derivative of tan⁻¹(5x)?

An integral that gives an inverse tangent

Evaluate the integral of 1/(9 + x²) from 0 to 3. Here a² = 9, so a = 3, and the list gives (1/a) tan⁻¹(x/a):

\int_0^3\frac{1}{9+x^2}\ dx=\left[\frac{1}{3}\tan^{-1}\frac{x}{3}\right]_0^3=\frac{1}{3}\tan^{-1}1-0=\frac{\pi}{12}

The binomial theorem

Find the coefficient of x³ in the expansion of (2 + x)⁵. The term in x³ has r = 3, with a = 2 and b = x:

\binom{5}{3}\times2^{2}\times x^3=10\times4\times x^3=40x^3

So the coefficient is 40. The power of 2 is n − r = 2, which is the slip students make most often. Check the full expansion in the Binomial Expansion Calculator.

De Moivre's theorem

Evaluate (1 + i)⁸. In polar form, 1 + i has r = √2 and θ = π/4, so

(1+i)^8=(\sqrt{2})^8\left(\cos\frac{8\pi}{4}+i\sin\frac{8\pi}{4}\right)=16(\cos2\pi+i\sin2\pi)=16

How the Advanced Higher list compares with Higher

The Higher list prints circle equations, the scalar product, the addition and double angle formulae, and the derivatives and integrals of sin ax and cos ax. The Advanced Higher list prints none of those. Read the Higher Maths formula sheet and keep it in your revision, because Advanced Higher questions use it without a reminder.

Exam answers

The answers and a worked mark scheme come with today's paper once it is marked. Sit the paper with the sheet open beside you, mark it, and the worked answers show which line each question wanted. We do not publish official or past paper answers.

Use the sheet with today's paper

Uni-Sine publishes a free daily paper for Advanced Higher Maths. It is a short timed paper of 8 questions worth 25 to 35 marks, not a full SQA paper and not set by SQA. Many schools set a longer prelim in winter: see our prelims guide. Open the sheet, set a timer, and write the line you use at the start of each answer.

Common mistakes

  • Forgetting the chain rule. The table gives the derivative of tan⁻¹ x, not of tan⁻¹ 3x. Multiply by the inside derivative.
  • Mixing up a² − x² and a² + x². One integral gives inverse sine, the other inverse tangent with a factor of 1/a.
  • Using the wrong power in the binomial theorem. The power of a is n − r.
  • Treating the list as a method. It does not say when to use integration by parts or partial fractions.
  • Using degrees in de Moivre. Keep the argument in radians unless the question uses degrees.

Common questions

Do you get a formula sheet in Advanced Higher Maths?
Yes. SQA prints a formulae list at the front of both Advanced Higher Mathematics papers.
What is on the Advanced Higher Maths formulae list?
Standard derivatives and integrals, the sums of arithmetic and geometric series and of r, r² and r³, the binomial theorem, the Maclaurin expansion, de Moivre's theorem, the vector product and the rotation matrix.
Are the Higher formulae on the Advanced Higher list?
No. The circle equations, scalar product and trigonometric formulae from Higher are not repeated.
Is integration by parts on the Advanced Higher formula sheet?
No. The list has standard results only, not the methods of integration.
Is the sum to infinity of a geometric series given?
No. Only the finite sum, S_n, is printed.

Revision cards

Say the answer first, then flip the card to check it.

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