A formula booklet is provided in every IB Analysis and Approaches paper, so you do not need to recite everything. This sheet collects the results that are worth knowing fluently for speed and for the paper without a calculator, such as exact values, rules for logs and powers, and standard derivatives and integrals. Use it alongside the booklet and learn to find the rest there quickly. Some items are for HL only.
| Formula | Meaning, symbols and units | When you use it |
|---|
| \(a^m\times a^n=a^{m+n}\) | Multiplying powers with the same base; \(m,n\) are exponents | Simplifying expressions without a calculator |
| \(\dfrac{a^m}{a^n}=a^{m-n}\) | Dividing powers with the same base | Simplifying and solving exponential equations |
| \((a^m)^n=a^{mn}\) | Power of a power | Rewriting both sides of an equation with a common base |
| \(a^{-n}=\dfrac{1}{a^n},\quad a^{1/n}=\sqrt[n]{a}\) | Negative and fractional exponents | Converting roots to powers before differentiating |
| \(\log_a x=y\iff a^y=x\) | Logarithm as the inverse of an exponential, \(a>0,\ a\ne1\) | Solving \(a^x=b\) exactly |
| \(\log_a(xy)=\log_a x+\log_a y\) | Log of a product | Combining or splitting logarithms |
| \(\log_a\dfrac{x}{y}=\log_a x-\log_a y\) | Log of a quotient | Simplifying log equations |
| \(\log_a x^n=n\log_a x\) | Log of a power | Bringing an unknown exponent down |
| \(\log_a a=1,\quad \log_a 1=0\) | Two basic log values | Simplifying |
| \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\) | Roots of \(ax^2+bx+c=0\) | Solving quadratics that do not factorise |
| \(\Delta=b^2-4ac\) | Discriminant: positive gives two roots, zero one, negative none | Counting roots, tangency conditions |
| \(x_1+x_2=-\dfrac ba,\quad x_1x_2=\dfrac ca\) | Sum and product of the roots of a quadratic | Finding unknowns without solving |
| \(a^2-b^2=(a-b)(a+b)\) | Difference of two squares | Factorising, rationalising |
| \(|x|=a\iff x=\pm a\) | Modulus equation, \(a\ge0\) | Solving equations containing absolute values |
| Formula | Meaning, symbols and units | When you use it |
|---|
| \(u_n=u_1+(n-1)d\) | \(n\)th term of an arithmetic sequence; \(d\) is the common difference | Finding a particular term |
| \(S_n=\dfrac n2\,(2u_1+(n-1)d)\) | Sum of the first \(n\) terms of an arithmetic series | Adding a run of equally spaced terms |
| \(u_n=u_1r^{\,n-1}\) | \(n\)th term of a geometric sequence; \(r\) is the common ratio | Growth and decay problems |
| \(S_n=\dfrac{u_1(r^n-1)}{r-1}\) | Sum of the first \(n\) terms of a geometric series, \(r\ne1\) | Totals of geometric terms |
| \(S_\infty=\dfrac{u_1}{1-r},\ |r|<1\) | Sum to infinity of a convergent geometric series | Recurring decimals, infinite series |
| \(\binom nr=\dfrac{n!}{r!\,(n-r)!}\) | Number of ways to choose \(r\) from \(n\) | Binomial expansion and probability |
| \((a+b)^n=\sum_{r=0}^{n}\binom nr a^{n-r}b^r\) | Binomial expansion for positive integer \(n\) | Expanding brackets, finding one term |
| \(A=P\left(1+\dfrac{r}{100k}\right)^{kn}\) | Compound growth: \(P\) start, \(r\)% yearly rate, \(k\) periods per year, \(n\) years | Money and population models |
| Formula | Meaning, symbols and units | When you use it |
|---|
| \((f\circ g)(x)=f(g(x))\) | Composite function: apply \(g\) first | Combining functions |
| \(f(f^{-1}(x))=x\) | Inverse function undoes \(f\); its graph reflects in \(y=x\) | Finding and checking inverses |
| \(y=af(b(x-c))+d\) | Stretch by \(a\) vertically and \(1/b\) horizontally, shift \(c\) right and \(d\) up | Describing graph transformations |
| \(y=a(x-h)^2+k\) | Vertex form, with vertex at \((h,k)\) | Finding the turning point of a quadratic |
| \(x=-\dfrac{b}{2a}\) | Axis of symmetry of \(y=ax^2+bx+c\) | Locating the vertex |
| \(y=\dfrac{ax+b}{cx+d}\) | Rational function: vertical asymptote \(x=-d/c\), horizontal asymptote \(y=a/c\) | Sketching reciprocal graphs |
| \(m=\dfrac{y_2-y_1}{x_2-x_1}\) | Gradient between two points | Equation of a straight line |
| \(m_1m_2=-1\) | Gradients of perpendicular lines | Normals and perpendicular bisectors |
| Formula | Meaning, symbols and units | When you use it |
|---|
| \(\sin^2\theta+\cos^2\theta=1\) | Fundamental identity | Converting between sine and cosine |
| \(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\) | Tangent as a ratio | Simplifying and solving |
| \(\sin2\theta=2\sin\theta\cos\theta\) | Double angle for sine | Solving equations, integrating |
| \(\cos2\theta=\cos^2\theta-\sin^2\theta=2\cos^2\theta-1=1-2\sin^2\theta\) | Three forms of the double angle for cosine | Choosing the form that suits the problem |
| \(s=r\theta,\quad A=\tfrac12r^2\theta\) | Arc length and sector area, \(\theta\) in radians, \(r\) in length units | Circle problems |
| \(180^\circ=\pi\text{ rad}\) | Degree and radian link | Converting between angle units |
| \(\sin\frac\pi6=\tfrac12,\ \cos\frac\pi6=\tfrac{\sqrt3}2,\ \tan\frac\pi6=\tfrac1{\sqrt3}\) | Exact values for 30° | Paper without a calculator |
| \(\sin\frac\pi4=\cos\frac\pi4=\tfrac{\sqrt2}2,\ \tan\frac\pi4=1\) | Exact values for 45° | Paper without a calculator |
| \(\sin\frac\pi3=\tfrac{\sqrt3}2,\ \cos\frac\pi3=\tfrac12,\ \tan\frac\pi3=\sqrt3\) | Exact values for 60° | Paper without a calculator |
| \(\sin(\pi-\theta)=\sin\theta,\ \cos(\pi-\theta)=-\cos\theta\) | Supplementary angle relationships | Finding the second solution |
| \(\sin\left(\tfrac\pi2-\theta\right)=\cos\theta\) | Complementary angle relationship | Rewriting expressions |
| \(\text{Area}=\tfrac12ab\sin C\) | Triangle area from two sides \(a,b\) and the angle \(C\) between them | Triangles without a known height |
| Formula | Meaning, symbols and units | When you use it |
|---|
| \(\dfrac{d}{dx}x^n=nx^{n-1}\) | Power rule, any real \(n\) | Differentiating polynomials and roots |
| \(\dfrac{d}{dx}e^x=e^x,\quad \dfrac{d}{dx}\ln x=\dfrac1x\) | Exponential and natural logarithm | Rates of growth, optimisation |
| \(\dfrac{d}{dx}\sin x=\cos x,\quad \dfrac{d}{dx}\cos x=-\sin x\) | Sine and cosine, \(x\) in radians | Oscillating models |
| \(\dfrac{dy}{dx}=\dfrac{dy}{du}\times\dfrac{du}{dx}\) | Chain rule | Differentiating composite functions |
| \(\dfrac{d}{dx}(uv)=u\dfrac{dv}{dx}+v\dfrac{du}{dx}\) | Product rule | Products of functions |
| \(\dfrac{d}{dx}\dfrac uv=\dfrac{v\,u\prime-u\,v\prime}{v^2}\) | Quotient rule | Ratios of functions |
| \(\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C,\ n\ne-1\) | Power rule for integration; \(C\) is the constant | Antiderivatives of polynomials |
| \(\int\dfrac1x\,dx=\ln|x|+C\) | Integral of the reciprocal | Areas, differential equations |
| \(\int e^x\,dx=e^x+C\) | Integral of the exponential | Growth models |
| \(A=\int_a^b y\,dx\) | Area under a curve above the \(x\)-axis between \(x=a\) and \(x=b\) | Areas and accumulated change |
| \(f\prime(x)=0,\quad f\prime\prime(x)\) | Stationary points; second derivative positive means minimum, negative maximum | Optimisation |
| \(v=\dfrac{ds}{dt},\quad a=\dfrac{dv}{dt}\) | Velocity and acceleration from displacement \(s\) and time \(t\) | Kinematics |
| \(\text{distance}=\int_{t_1}^{t_2}|v|\,dt\) | Total distance travelled from velocity \(v\) | Motion with direction changes |
| Formula | Meaning, symbols and units | When you use it |
|---|
| \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\) | Probability of A or B | Overlapping events |
| \(P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}\) | Conditional probability | Tree diagrams, two-way tables |
| \(P(A\cap B)=P(A)P(B)\) | Holds only for independent events | Testing independence |
| \(P(X=r)=\dbinom nr p^r(1-p)^{n-r}\) | Binomial distribution: \(n\) trials, success probability \(p\) | Counting successes |
| \(E(X)=np,\quad \mathrm{Var}(X)=np(1-p)\) | Mean and variance of a binomial variable | Describing a binomial distribution |
| \(z=\dfrac{x-\mu}{\sigma}\) | Standardised normal value; \(\mu\) mean, \(\sigma\) standard deviation | Comparing values, using tables or inverse normal |
| \(\mathrm{IQR}=Q_3-Q_1\) | Spread of the middle half of the data | Spotting outliers |
| \(\text{outlier if }xQ_3+1.5\,\mathrm{IQR}\) | Standard outlier test | Box plots |
| \(E(X)=\sum xP(X=x)\) | Expected value of a discrete variable | Games and expected gain |
| Formula | Meaning, symbols and units | When you use it |
|---|
| \(|\mathbf a|=\sqrt{a_1^2+a_2^2+a_3^2}\) | Magnitude of a vector (HL) | Distances and unit vectors |
| \(\mathbf a\cdot\mathbf b=|\mathbf a||\mathbf b|\cos\theta\) | Scalar product (HL); \(\theta\) is the angle between the vectors | Finding angles, testing perpendicularity |
| \(\mathbf r=\mathbf a+\lambda\mathbf d\) | Line through point \(\mathbf a\) with direction \(\mathbf d\) (HL) | Intersections and angles between lines |
| \(|\mathbf a\times\mathbf b|=|\mathbf a||\mathbf b|\sin\theta\) | Magnitude of the vector product (HL); also the area of the parallelogram on \(\mathbf a\) and \(\mathbf b\) | Areas of triangles and parallelograms in 3D |