The formula booklet
Every formula the IB gives you in the examinations for Analysis and Approaches, in the order the booklet itself uses, with a note from me wherever the printed line is not the whole story. Version 1.3 is the current one, and it is the booklet you will be handed in every paper — including Paper 1, where you have no calculator.
✅What this page is
- The formulas listed in the IB’s own booklet for this course — version 1.3, first examinations 2021 — restated here so you can search them on a phone.
- Not the booklet itself. There is no cover page and no IB branding here: the booklet is an IB publication and this is our reference sheet. Work from the clean copy you are issued in the exam, and ask your teacher or me for the official PDF.
- A formula is a fact, and nobody owns \( u_n = u_1 + (n-1)d \). What is ours on this page is the commentary: notes in this grey are mine, not the IB’s.
- Still current: this version is the one in use for examinations up to and including November 2028. The revised DP mathematics courses are first taught in 2027 and first assessed in May 2029, and they will bring a new booklet.
🧠How to use it — and what it will not do for you
Students who lose marks to the booklet almost never lose them because a formula was missing. They lose them because they did not know a line existed, or because they knew it existed and stopped thinking once they had copied it.
Practise with it open, from day one. The booklet in the exam is clean, so anything you have written in the margin of yours you will have to carry in your head. Knowing which page a formula is on is worth more under time pressure than knowing the formula.
The codes are the index. Each line is tagged with the sub-topic it belongs to — \(\text{SL }1.2\) is sequences, \(\text{AHL }5.19\) is Maclaurin series. When you cannot find something, work out which sub-topic it belongs to and look there; that is how the booklet is organised, not alphabetically.
What is given tells you what is examined. A formula that is printed is one you are never asked to derive; a formula that is absent is one you are expected to know, build, or get from your GDC. The list at the bottom of this page is the one worth reading twice.
🔢Prior learning · SL and HL
—
Area of a parallelogram
\[ A = bh \]where \(b\) is the base, \(h\) is the height
Area of a triangle
\[ A = \tfrac{1}{2}(bh) \]where \(b\) is the base, \(h\) is the height
Area of a trapezoid
\[ A = \tfrac{1}{2}(a+b)h \]where \(a\) and \(b\) are the parallel sides, \(h\) is the height
Area of a circle
\[ A = \pi r^{2} \]where \(r\) is the radius
Circumference of a circle
\[ C = 2\pi r \]where \(r\) is the radius
Volume of a cuboid
\[ V = lwh \]where \(l\) is the length, \(w\) is the width, \(h\) is the height
Volume of a cylinder
\[ V = \pi r^{2}h \]where \(r\) is the radius, \(h\) is the height
Volume of a prism
\[ V = Ah \]where \(A\) is the area of cross-section, \(h\) is the height
Area of the curved surface of a cylinder
\[ A = 2\pi rh \]where \(r\) is the radius, \(h\) is the height
Distance between two points \((x_1,\,y_1)\) and \((x_2,\,y_2)\)
\[ d = \sqrt{(x_1-x_2)^{2} + (y_1-y_2)^{2}} \]
Coordinates of the midpoint of a line segment with endpoints \((x_1,\,y_1)\) and \((x_2,\,y_2)\)
\[ \left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right) \]
🔢Topic 1: Number and algebra · SL and HL
SL
1.2
1.2
The \(n\)th term of an arithmetic sequence
\[ u_n = u_1 + (n-1)d \]
The sum of \(n\) terms of an arithmetic sequence
\[ S_n = \dfrac{n}{2}\bigl(2u_1 + (n-1)d\bigr)\,;\quad S_n = \dfrac{n}{2}(u_1 + u_n) \]
Two forms of the same sum. The second is the faster one whenever you already know the last term \(u_n\) — which is most “sum of the first \(n\) terms” questions.
SL
1.3
1.3
The \(n\)th term of a geometric sequence
\[ u_n = u_1 r^{\,n-1} \]
The sum of \(n\) terms of a finite geometric sequence
\[ S_n = \dfrac{u_1(r^{n}-1)}{r-1} = \dfrac{u_1(1-r^{n})}{1-r},\quad r \ne 1 \]
SL
1.4
1.4
Compound interest
\[ FV = PV \times \left(1 + \dfrac{r}{100k}\right)^{kn} \]where \(FV\) is the future value, \(PV\) is the present value, \(n\) is the number of years, \(k\) is the number of compounding periods per year, \(r\%\) is the nominal annual rate of interest
The \(100\) is already in the formula, so \(r\) goes in as a percentage (\(3.5\), not \(0.035\)). \(k\) is compounding periods per year and \(n\) is years, not periods.
SL
1.5
1.5
Exponents and logarithms
\[ a^{x} = b \iff x = \log_a b \]where \(a > 0\), \(b > 0\), \(a \ne 1\)
SL
1.7
1.7
Exponents and logarithms
\[ \log_a xy = \log_a x + \log_a y \]
\[ \log_a \dfrac{x}{y} = \log_a x - \log_a y \]
\[ \log_a x^{m} = m\log_a x \]
\[ \log_a x = \dfrac{\log_b x}{\log_b a} \]
The change-of-base line is the one students forget they are given. It is what turns \(\log_3 40\) into something a calculator will evaluate.
SL
1.8
1.8
The sum of an infinite geometric sequence
\[ S_\infty = \dfrac{u_1}{1-r},\quad |r| < 1 \]
The condition is part of the formula. A question asking for which values of \(x\) a sum to infinity exists is asking you to solve \(|r| < 1\).
SL
1.9
1.9
Binomial theorem, \(n \in \mathbb{N}\)
\[ (a+b)^{n} = a^{n} + {}^{n}\text{C}_1\,a^{n-1}b + \dots + {}^{n}\text{C}_r\,a^{n-r}b^{r} + \dots + b^{n} \]
\[ {}^{n}\text{C}_r = \dfrac{n!}{r!\,(n-r)!} \]
Count the term you want before substituting: the term in \(b^{r}\) is the \((r+1)\)th term, which is where most of the lost marks in this sub-topic come from.
🔢Topic 1: Number and algebra · HL only
AHL
1.10
1.10
Combinations
\[ {}^{n}\text{C}_r = \dfrac{n!}{r!\,(n-r)!} \]
Permutations
\[ {}^{n}\text{P}_r = \dfrac{n!}{(n-r)!} \]
Both are printed, so the only thing being marked is which one the context needs — whether order matters.
Extension of binomial theorem, \(n \in \mathbb{Q}\)
\[ (a+b)^{n} = a^{n}\left(1 + n\left(\dfrac{b}{a}\right) + \dfrac{n(n-1)}{2!}\left(\dfrac{b}{a}\right)^{2} + \dots\right) \]
Note the form: \(a\) factored out, the expansion in \(b/a\). It is valid only for \(\left|\dfrac{b}{a}\right| < 1\), and that condition is not printed — you are expected to state it.
AHL
1.12
1.12
Complex numbers
\[ z = a + b\mathrm{i} \]
AHL
1.13
1.13
Modulus-argument (polar) and exponential (Euler) form
\[ z = r(\cos\theta + \mathrm{i}\sin\theta) = r\mathrm{e}^{\mathrm{i}\theta} = r\operatorname{cis}\theta \]
AHL
1.14
1.14
De Moivre's theorem
\[ [r(\cos\theta + \mathrm{i}\sin\theta)]^{n} = r^{n}(\cos n\theta + \mathrm{i}\sin n\theta) = r^{n}\mathrm{e}^{\mathrm{i}n\theta} = r^{n}\operatorname{cis} n\theta \]
Used with \(n = \tfrac{1}{k}\) for roots. That the \(k\) roots come from adding \(2\pi\) to the argument before dividing is not shown anywhere in the booklet.
🔢Topic 2: Functions · SL and HL
SL
2.1
2.1
Equations of a straight line
\[ y = mx + c\,;\quad ax + by + d = 0\,;\quad y - y_1 = m(x-x_1) \]
Gradient formula
\[ m = \dfrac{y_2-y_1}{x_2-x_1} \]
SL
2.6
2.6
Axis of symmetry of the graph of a quadratic function
\[ f(x) = ax^{2} + bx + c \Rightarrow \text{axis of symmetry is } x = -\dfrac{b}{2a} \]
The same line gives you the vertex: substitute that \(x\) back in for the \(y\)-coordinate.
SL
2.7
2.7
Solutions of a quadratic equation
\[ ax^{2}+bx+c = 0 \Rightarrow x = \dfrac{-b \pm \sqrt{b^{2}-4ac}}{2a},\quad a \ne 0 \]
Discriminant
\[ \Delta = b^{2} - 4ac \]
The expression is given but not what it tells you: \(\Delta > 0\) two roots, \(\Delta = 0\) one repeated root, \(\Delta < 0\) none. Learn that, not this.
SL
2.9
2.9
Exponential and logarithmic functions
\[ a^{x} = \mathrm{e}^{x\ln a}\,;\quad \log_a a^{x} = x = a^{\log_a x} \]where \(a,\,x > 0\), \(a \ne 1\)
🔢Topic 2: Functions · HL only
AHL
2.12
2.12
Sum and product of the roots of polynomial equations of the form \(\sum\limits_{r=0}^{n} a_r x^{r} = 0\)
\[ \text{sum is } \dfrac{-a_{n-1}}{a_n}\,;\quad \text{product is } \dfrac{(-1)^{n}a_0}{a_n} \]
🔢Topic 3: Geometry and trigonometry · SL and HL
SL
3.1
3.1
Distance between two points \((x_1,y_1,z_1)\) and \((x_2,y_2,z_2)\)
\[ d = \sqrt{(x_1-x_2)^{2} + (y_1-y_2)^{2} + (z_1-z_2)^{2}} \]
Coordinates of the midpoint of a line segment with endpoints \((x_1,y_1,z_1)\) and \((x_2,y_2,z_2)\)
\[ \left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2},\ \dfrac{z_1+z_2}{2}\right) \]
Volume of a right-pyramid
\[ V = \tfrac{1}{3}Ah \]where \(A\) is the area of the base, \(h\) is the height
Volume of a right cone
\[ V = \tfrac{1}{3}\pi r^{2}h \]where \(r\) is the radius, \(h\) is the height
Area of the curved surface of a cone
\[ A = \pi r l \]where \(r\) is the radius, \(l\) is the slant height
Volume of a sphere
\[ V = \tfrac{4}{3}\pi r^{3} \]where \(r\) is the radius
Surface area of a sphere
\[ A = 4\pi r^{2} \]where \(r\) is the radius
SL
3.2
3.2
Sine rule
\[ \dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} \]
Nothing in the booklet warns you about the ambiguous case: when you use the sine rule to find an angle, check whether the obtuse solution also fits the triangle.
Cosine rule
\[ c^{2} = a^{2} + b^{2} - 2ab\cos C\,;\quad \cos C = \dfrac{a^{2}+b^{2}-c^{2}}{2ab} \]
Area of a triangle
\[ A = \tfrac{1}{2}ab\sin C \]
The angle must be the one between the two sides. Spotting that is the whole content of most “find the area” questions.
SL
3.4
3.4
Length of an arc
\[ l = r\theta \]where \(r\) is the radius, \(\theta\) is the angle measured in radians
Area of a sector
\[ A = \tfrac{1}{2}r^{2}\theta \]where \(r\) is the radius, \(\theta\) is the angle measured in radians
Radians only — these are the AA forms. The degree versions (\(\tfrac{\theta}{360}\times 2\pi r\)) are in the AI booklet, not yours.
SL
3.5
3.5
Identity for \(\tan\theta\)
\[ \tan\theta = \dfrac{\sin\theta}{\cos\theta} \]
SL
3.6
3.6
Pythagorean identity
\[ \cos^{2}\theta + \sin^{2}\theta = 1 \]
Double angle identities
\[ \sin 2\theta = 2\sin\theta\cos\theta \]
\[ \cos 2\theta = \cos^{2}\theta - \sin^{2}\theta = 2\cos^{2}\theta - 1 = 1 - 2\sin^{2}\theta \]
Three forms of \(\cos 2\theta\) are printed for a reason: choose the one that leaves the equation in a single trigonometric function.
🔢Topic 3: Geometry and trigonometry · HL only
AHL
3.9
3.9
Reciprocal trigonometric identities
\[ \sec\theta = \dfrac{1}{\cos\theta}\,;\quad \operatorname{cosec}\theta = \dfrac{1}{\sin\theta} \]
Pythagorean identities
\[ 1 + \tan^{2}\theta = \sec^{2}\theta \]
\[ 1 + \cot^{2}\theta = \operatorname{cosec}^{2}\theta \]
AHL
3.10
3.10
Compound angle identities
\[ \sin(A \pm B) = \sin A\cos B \pm \cos A\sin B \]
\[ \cos(A \pm B) = \cos A\cos B \mp \sin A\sin B \]
\[ \tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A\tan B} \]
Watch the \(\mp\) in the cosine and tangent lines: the lower sign there goes with the \(+\) on the left.
Double angle identity for \(\tan\)
\[ \tan 2\theta = \dfrac{2\tan\theta}{1-\tan^{2}\theta} \]
AHL
3.12
3.12
Magnitude of a vector
\[ |\boldsymbol{v}| = \sqrt{v_1^{2}+v_2^{2}+v_3^{2}}, \quad \text{where } \boldsymbol{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix} \]
AHL
3.13
3.13
Scalar product
\[ \boldsymbol{v}\cdot\boldsymbol{w} = v_1w_1 + v_2w_2 + v_3w_3, \quad \text{where } \boldsymbol{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix},\ \boldsymbol{w} = \begin{pmatrix} w_1 \\ w_2 \\ w_3 \end{pmatrix} \]
\[ \boldsymbol{v}\cdot\boldsymbol{w} = |\boldsymbol{v}||\boldsymbol{w}|\cos\theta \]where \(\theta\) is the angle between \(\boldsymbol{v}\) and \(\boldsymbol{w}\)
Angle between two vectors
\[ \cos\theta = \dfrac{v_1w_1+v_2w_2+v_3w_3}{|\boldsymbol{v}||\boldsymbol{w}|} \]
That perpendicular vectors are the case \(\boldsymbol{v}\cdot\boldsymbol{w} = 0\) is nowhere in the booklet, and it is what most vector questions turn on.
AHL
3.14
3.14
Vector equation of a line
\[ \boldsymbol{r} = \boldsymbol{a} + \lambda\boldsymbol{b} \]
Parametric form of the equation of a line
\[ x = x_0 + \lambda l,\quad y = y_0 + \lambda m,\quad z = z_0 + \lambda n \]
Cartesian equations of a line
\[ \dfrac{x-x_0}{l} = \dfrac{y-y_0}{m} = \dfrac{z-z_0}{n} \]
AHL
3.16
3.16
Vector product
\[ \boldsymbol{v}\times\boldsymbol{w} = \begin{pmatrix} v_2w_3 - v_3w_2 \\ v_3w_1 - v_1w_3 \\ v_1w_2 - v_2w_1 \end{pmatrix}, \quad \text{where } \boldsymbol{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix},\ \boldsymbol{w} = \begin{pmatrix} w_1 \\ w_2 \\ w_3 \end{pmatrix} \]
Copy the middle row carefully: it is \(v_3w_1 - v_1w_3\), the one that looks “backwards”.
\[ |\boldsymbol{v}\times\boldsymbol{w}| = |\boldsymbol{v}||\boldsymbol{w}|\sin\theta \]where \(\theta\) is the angle between \(\boldsymbol{v}\) and \(\boldsymbol{w}\)
Area of a parallelogram
\[ A = |\boldsymbol{v}\times\boldsymbol{w}| \]where \(\boldsymbol{v}\) and \(\boldsymbol{w}\) form two adjacent sides of a parallelogram
AHL
3.17
3.17
Vector equation of a plane
\[ \boldsymbol{r} = \boldsymbol{a} + \lambda\boldsymbol{b} + \mu\boldsymbol{c} \]
Equation of a plane (using the normal vector)
\[ \boldsymbol{r}\cdot\boldsymbol{n} = \boldsymbol{a}\cdot\boldsymbol{n} \]
Cartesian equation of a plane
\[ ax + by + cz = d \]
The three forms are the three things a plane question can hand you. Getting from the vector equation to the Cartesian one means finding \(\boldsymbol{n}\) with a vector product first.
🔢Topic 4: Statistics and probability · SL and HL
SL
4.2
4.2
Interquartile range
\[ \text{IQR} = Q_3 - Q_1 \]
SL
4.3
4.3
Mean, \(\overline{x}\), of a set of data
\[ \overline{x} = \dfrac{\sum\limits_{i=1}^{k} f_i x_i}{n}, \quad \text{where } n = \sum_{i=1}^{k} f_i \]
SL
4.5
4.5
Probability of an event \(A\)
\[ \text{P}(A) = \dfrac{n(A)}{n(U)} \]
Complementary events
\[ \text{P}(A) + \text{P}(A') = 1 \]
SL
4.6
4.6
Combined events
\[ \text{P}(A \cup B) = \text{P}(A) + \text{P}(B) - \text{P}(A \cap B) \]
Mutually exclusive events
\[ \text{P}(A \cup B) = \text{P}(A) + \text{P}(B) \]
Conditional probability
\[ \text{P}(A \mid B) = \dfrac{\text{P}(A \cap B)}{\text{P}(B)} \]
Independent events
\[ \text{P}(A \cap B) = \text{P}(A)\text{P}(B) \]
All four sit under SL 4.6. The two you have to choose between are the last two: multiplying probabilities is only legal once independence is given or shown.
SL
4.7
4.7
Expected value of a discrete random variable \(X\)
\[ \text{E}(X) = \sum x\,\text{P}(X = x) \]
SL
4.8
4.8
Binomial distribution
\[ X \sim \text{B}(n,\,p) \]
Mean
\[ \text{E}(X) = np \]
Variance
\[ \operatorname{Var}(X) = np(1-p) \]
The probability itself, \({}^{n}\text{C}_x\,p^{x}(1-p)^{n-x}\), is not in the booklet — your GDC is expected to produce it.
SL
4.12
4.12
Standardized normal variable
\[ z = \dfrac{x-\mu}{\sigma} \]
🔢Topic 4: Statistics and probability · HL only
AHL
4.13
4.13
Bayes' theorem
\[ \text{P}(B \mid A) = \dfrac{\text{P}(B)\text{P}(A \mid B)}{\text{P}(B)\text{P}(A \mid B) + \text{P}(B')\text{P}(A \mid B')} \]
\[ \text{P}(B_i \mid A) = \dfrac{\text{P}(B_i)\text{P}(A \mid B_i)}{\text{P}(B_1)\text{P}(A \mid B_1) + \text{P}(B_2)\text{P}(A \mid B_2) + \text{P}(B_3)\text{P}(A \mid B_3)} \]
The second line stops at three partitions because that is as far as the syllabus goes. A tree diagram reproduces either line and is much harder to mis-copy under pressure.
AHL
4.14
4.14
Variance \(\sigma^{2}\)
\[ \sigma^{2} = \dfrac{\sum\limits_{i=1}^{k} f_i(x_i-\mu)^{2}}{n} = \dfrac{\sum\limits_{i=1}^{k} f_i x_i^{2}}{n} - \mu^{2} \]
Standard deviation \(\sigma\)
\[ \sigma = \sqrt{\dfrac{\sum\limits_{i=1}^{k} f_i(x_i-\mu)^{2}}{n}} \]
Linear transformation of a single random variable
\[ \text{E}(aX+b) = a\text{E}(X) + b\,;\quad \operatorname{Var}(aX+b) = a^{2}\operatorname{Var}(X) \]
The \(b\) disappears from the variance: shifting data does not spread it. This is the line examiners most often report candidates ignoring.
Expected value of a continuous random variable \(X\)
\[ \text{E}(X) = \mu = \int_{-\infty}^{\infty} x f(x)\,\mathrm{d}x \]
Variance
\[ \operatorname{Var}(X) = \text{E}\bigl[(X-\mu)^{2}\bigr] = \text{E}(X^{2}) - [\text{E}(X)]^{2} \]
Variance of a discrete random variable \(X\)
\[ \operatorname{Var}(X) = \sum (x-\mu)^{2}\text{P}(X=x) = \sum x^{2}\text{P}(X=x) - \mu^{2} \]
Variance of a continuous random variable \(X\)
\[ \operatorname{Var}(X) = \int_{-\infty}^{\infty} (x-\mu)^{2} f(x)\,\mathrm{d}x = \int_{-\infty}^{\infty} x^{2} f(x)\,\mathrm{d}x - \mu^{2} \]
🔢Topic 5: Calculus · SL and HL
SL
5.3
5.3
Derivative of \(x^{n}\)
\[ f(x) = x^{n} \Rightarrow f'(x) = nx^{\,n-1} \]
SL
5.5
5.5
Integral of \(x^{n}\)
\[ \int x^{n}\,\mathrm{d}x = \dfrac{x^{\,n+1}}{n+1} + C,\quad n \ne -1 \]
Area between a curve \(y = f(x)\) and the \(x\)-axis, where \(f(x) > 0\)
\[ A = \int_a^b y\,\mathrm{d}x \]
Note the condition \(f(x) > 0\). Where the curve dips below the axis you need the SL 5.11 line instead.
SL
5.6
5.6
Derivative of \(\sin x\)
\[ f(x) = \sin x \Rightarrow f'(x) = \cos x \]
Derivative of \(\cos x\)
\[ f(x) = \cos x \Rightarrow f'(x) = -\sin x \]
Derivative of \(\mathrm{e}^{x}\)
\[ f(x) = \mathrm{e}^{x} \Rightarrow f'(x) = \mathrm{e}^{x} \]
Derivative of \(\ln x\)
\[ f(x) = \ln x \Rightarrow f'(x) = \dfrac{1}{x} \]
Chain rule
\[ y = g(u), \text{ where } u = f(x) \Rightarrow \dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{\mathrm{d}y}{\mathrm{d}u} \times \dfrac{\mathrm{d}u}{\mathrm{d}x} \]
Product rule
\[ y = uv \Rightarrow \dfrac{\mathrm{d}y}{\mathrm{d}x} = u\dfrac{\mathrm{d}v}{\mathrm{d}x} + v\dfrac{\mathrm{d}u}{\mathrm{d}x} \]
Quotient rule
\[ y = \dfrac{u}{v} \Rightarrow \dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{v\dfrac{\mathrm{d}u}{\mathrm{d}x} - u\dfrac{\mathrm{d}v}{\mathrm{d}x}}{v^{2}} \]
The derivatives of \(\tan x\), \(a^{x}\) and \(\log_a x\) are not here at SL — they are AHL 5.15.
SL
5.9
5.9
Acceleration
\[ a = \dfrac{\mathrm{d}v}{\mathrm{d}t} = \dfrac{\mathrm{d}^{2}s}{\mathrm{d}t^{2}} \]
Distance travelled from \(t_1\) to \(t_2\)
\[ \text{distance} = \int_{t_1}^{t_2} |v(t)|\,\mathrm{d}t \]
Displacement from \(t_1\) to \(t_2\)
\[ \text{displacement} = \int_{t_1}^{t_2} v(t)\,\mathrm{d}t \]
The modulus signs are the entire difference between these two lines — and between full marks and half on the standard kinematics question.
SL
5.10
5.10
Standard integrals
\[ \int \dfrac{1}{x}\,\mathrm{d}x = \ln|x| + C \]
\[ \int \sin x\,\mathrm{d}x = -\cos x + C \]
\[ \int \cos x\,\mathrm{d}x = \sin x + C \]
\[ \int \mathrm{e}^{x}\,\mathrm{d}x = \mathrm{e}^{x} + C \]
SL
5.11
5.11
Area of region enclosed by a curve and \(x\)-axis
\[ A = \int_a^b |y|\,\mathrm{d}x \]
In practice: split the integral at every \(x\)-intercept and add the magnitudes. On Paper 2 a GDC will evaluate \(\int |y|\,\mathrm{d}x\) directly.
🔢Topic 5: Calculus · HL only
AHL
5.12
5.12
Derivative of \(f(x)\) from first principles
\[ y = f(x) \Rightarrow \dfrac{\mathrm{d}y}{\mathrm{d}x} = f'(x) = \lim_{h \to 0}\left(\dfrac{f(x+h) - f(x)}{h}\right) \]
AHL
5.15
5.15
Standard derivatives — \(\tan x\)
\[ f(x) = \tan x \Rightarrow f'(x) = \sec^{2}x \]
\(\sec x\)
\[ f(x) = \sec x \Rightarrow f'(x) = \sec x \tan x \]
\(\operatorname{cosec} x\)
\[ f(x) = \operatorname{cosec} x \Rightarrow f'(x) = -\operatorname{cosec} x \cot x \]
\(\cot x\)
\[ f(x) = \cot x \Rightarrow f'(x) = -\operatorname{cosec}^{2}x \]
\(a^{x}\)
\[ f(x) = a^{x} \Rightarrow f'(x) = a^{x}(\ln a) \]
\(\log_a x\)
\[ f(x) = \log_a x \Rightarrow f'(x) = \dfrac{1}{x\ln a} \]
\(\arcsin x\)
\[ f(x) = \arcsin x \Rightarrow f'(x) = \dfrac{1}{\sqrt{1-x^{2}}} \]
\(\arccos x\)
\[ f(x) = \arccos x \Rightarrow f'(x) = -\dfrac{1}{\sqrt{1-x^{2}}} \]
\(\arctan x\)
\[ f(x) = \arctan x \Rightarrow f'(x) = \dfrac{1}{1+x^{2}} \]
Nine derivatives, none of them provable in the time a question allows — which is exactly why they are given and why recognising which one you need is the marked skill.
AHL
5.15
5.15
Standard integrals
\[ \int a^{x}\,\mathrm{d}x = \dfrac{1}{\ln a}a^{x} + C \]
\[ \int \dfrac{1}{a^{2}+x^{2}}\,\mathrm{d}x = \dfrac{1}{a}\arctan\left(\dfrac{x}{a}\right) + C \]
\[ \int \dfrac{1}{\sqrt{a^{2}-x^{2}}}\,\mathrm{d}x = \arcsin\left(\dfrac{x}{a}\right) + C,\quad |x| < a \]
These two are the tell that a substitution is wanted. Recognising \(a^{2}+x^{2}\) or \(a^{2}-x^{2}\) under the integral is the skill; the antiderivative is handed to you.
AHL
5.16
5.16
Integration by parts
\[ \int u\dfrac{\mathrm{d}v}{\mathrm{d}x}\,\mathrm{d}x = uv - \int v\dfrac{\mathrm{d}u}{\mathrm{d}x}\,\mathrm{d}x \quad \text{or} \quad \int u\,\mathrm{d}v = uv - \int v\,\mathrm{d}u \]
What to choose for \(u\) is not in the booklet: take the factor that gets simpler when differentiated (a polynomial, or \(\ln x\)).
AHL
5.17
5.17
Area of region enclosed by a curve and \(y\)-axis
\[ A = \int_a^b |x|\,\mathrm{d}y \]
Volume of revolution about the \(x\) or \(y\)-axes
\[ V = \int_a^b \pi y^{2}\,\mathrm{d}x \quad \text{or} \quad V = \int_a^b \pi x^{2}\,\mathrm{d}y \]
The limits belong to the variable you are integrating with respect to — for the \(y\)-axis version they are \(y\)-values. That is where the marks go.
AHL
5.18
5.18
Euler's method
\[ y_{n+1} = y_n + h \times f(x_n,\,y_n)\,;\quad x_{n+1} = x_n + h \]where \(h\) is a constant (step length)
Integrating factor for \(y' + P(x)y = Q(x)\)
\[ \mathrm{e}^{\int P(x)\,\mathrm{d}x} \]
The factor is given, the method is not: multiply through, recognise the left side as the derivative of a product, then integrate once.
AHL
5.19
5.19
Maclaurin series
\[ f(x) = f(0) + x f'(0) + \dfrac{x^{2}}{2!}f''(0) + \dots \]
Maclaurin series for special functions
\[ \mathrm{e}^{x} = 1 + x + \dfrac{x^{2}}{2!} + \dots \]
\[ \ln(1+x) = x - \dfrac{x^{2}}{2} + \dfrac{x^{3}}{3} - \dots \]
\[ \sin x = x - \dfrac{x^{3}}{3!} + \dfrac{x^{5}}{5!} - \dots \]
\[ \cos x = 1 - \dfrac{x^{2}}{2!} + \dfrac{x^{4}}{4!} - \dots \]
\[ \arctan x = x - \dfrac{x^{3}}{3} + \dfrac{x^{5}}{5} - \dots \]
Five standard series are given, so most questions are about composing, differentiating or integrating them rather than deriving one from scratch.
⚠What is not in the booklet
These come up every session and none of them are printed. This is the part of the page to reread the night before.
- The sum and product rules for probability are there; the binomial probability \( {}^{n}\text{C}_x p^{x}(1-p)^{n-x} \) is not. Nor is any normal-distribution density. Both are GDC work.
- No \( \sin^{2}\theta = \tfrac{1}{2}(1 - \cos 2\theta) \). The half-angle rearrangements you need for integrating \( \sin^{2}x \) come from the \( \cos 2\theta \) line in SL 3.6, and you are expected to do that rearranging.
- No exact trigonometric values. \( \sin 60^\circ = \tfrac{\sqrt{3}}{2} \) and its relatives are not printed, and Paper 1 is the no-calculator paper. Learn the two triangles.
- No derivative of \( \tan x \) at SL — it is AHL 5.15. Nor any antiderivative of \( \tan x \), at either level.
- No conditions for convergence beyond \( |r| < 1 \) for a geometric series: the binomial extension at AHL 1.10 needs \( \left|\tfrac{b}{a}\right| < 1 \) and you must state it yourself.
- No definition of a logarithm in words, no laws of indices, no factor or remainder theorem, no partial fractions. All examinable, none printed.
- Nothing about the Exploration. Criteria, word count and the five marks for personal engagement live in the guide, not here.
✏Find it in the booklet
Six drills. The skill being practised is not algebra — it is deciding which line you need, and noticing when there isn’t one. Try each before opening the answer.
A Paper 1 question gives \( f(x) = \dfrac{2x+1}{x-3} \) and asks for \( f'(x) \). Which line of the booklet do you need, and what does it not give you?
The quotient rule, SL 5.6. What the booklet does not give you is the algebra afterwards: \( f'(x) = \dfrac{(x-3)(2) - (2x+1)(1)}{(x-3)^{2}} = \dfrac{-7}{(x-3)^{2}} \). Most of the marks are in that simplification, and in not mixing up which term is subtracted.
You need the area enclosed between \( y = x^{3} - 4x \) and the \( x \)-axis from \( x = -2 \) to \( x = 2 \). Is that SL 5.5 or SL 5.11, and why does it matter?
SL 5.11, \( A = \int_a^b |y| \,\mathrm{d}x \). SL 5.5 carries the condition \( f(x) > 0 \), and this curve is above the axis on \( (-2,\,0) \) and below it on \( (0,\,2) \). Integrating \( y \) straight through gives \( 0 \) — a correct integral and a wrong area. Split at \( x = 0 \) and add the magnitudes: \( 4 + 4 = 8 \).
A question asks you to find the value of \( \log_5 90 \) to three significant figures. Which line turns that into something your calculator will do?
The change-of-base line in SL 1.7: \( \log_a x = \dfrac{\log_b x}{\log_b a} \), so \( \log_5 90 = \dfrac{\ln 90}{\ln 5} = 2.80 \). Students routinely believe this one is not given.
HL. You are asked for the exact value of \( \sin \dfrac{7\pi}{12} \). How much help is the booklet?
Some, but not all. AHL 3.10 gives \( \sin(A+B) = \sin A\cos B + \cos A\sin B \), and \( \dfrac{7\pi}{12} = \dfrac{\pi}{3} + \dfrac{\pi}{4} \). What is not given is the exact values of \( \sin\dfrac{\pi}{3} \), \( \cos\dfrac{\pi}{4} \) and the rest — those you must know. Answer: \( \dfrac{\sqrt{6}+\sqrt{2}}{4} \).
\( X \sim \text{B}(20,\,0.35) \). The question asks for \( \text{P}(X = 8) \) and then for the standard deviation of \( X \). Which part is in the booklet?
Only the second. SL 4.8 gives \( \operatorname{Var}(X) = np(1-p) = 20(0.35)(0.65) = 4.55 \), so \( \sigma = \sqrt{4.55} = 2.13 \). \( \text{P}(X = 8) \) has no printed formula: it is binomial pdf on the GDC (\( 0.1614 \)). Write down the distribution and the parameters as your method line.
HL. Find \( \displaystyle\int \dfrac{1}{\sqrt{9-x^{2}}}\,\mathrm{d}x \). Where in the booklet, and what do you have to spot?
AHL 5.15, standard integrals: \( \displaystyle\int \dfrac{1}{\sqrt{a^{2}-x^{2}}}\,\mathrm{d}x = \arcsin\left(\dfrac{x}{a}\right) + C \). The spotting is \( 9 = 3^{2} \), so \( a = 3 \) and the answer is \( \arcsin\left(\dfrac{x}{3}\right) + C \), valid for \( |x| < 3 \). Recognising the shape of the denominator is the marked skill; the antiderivative is given.
🔗Where to get the official booklet
These are external resources, not mine. If one stops working, tell me and everything above it on this page still stands.
- My IB → Programme Resource Centre — the authoritative copy of Mathematics: analysis and approaches formula booklet, version 1.3. Your school’s IB login reaches it; ask me if you cannot.
- Your teacher — schools are licensed to give you a copy, and most will print one. That is the simplest route and the one I would take.
- ibo.org → Diploma Programme → Mathematics — the public subject pages, for the guides, subject briefs and specimen material.