Knowledge organisers

Calculus: IB Maths AA SL knowledge organiser

Everything to know about calculus on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Derivative: f′(x) gives the gradient of the curve y = f(x) at each point: the rate of change of y with x.
Stationary point: A point where f′(x) = 0: a local maximum, a local minimum or a point of inflection with zero gradient.
Definite integral: ∫ₐᵇ f(x) dx gives the signed area between the curve and the x-axis from x = a to x = b.
Chain rule: To differentiate a function of a function: dy/dx = dy/du × du/dx.

Key formulas

Formulas marked with a label are given in the exam (we only say so where our formula sheet confirms it). Learn the rest.

Power ruleIn the formula booklet	\(f(x)=x^n\Rightarrow f'(x)=nx^{n-1}\)
Standard derivativesIn the formula booklet	\((\sin x)'=\cos x,\) \((\cos x)'=-\sin x,\) \((e^x)'=e^x,\) \((\ln x)'=\tfrac1x\)
Chain ruleIn the formula booklet	\(y=g(u),\ u=f(x)\Rightarrow\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\)
Product ruleIn the formula booklet	\(\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}\)
Quotient ruleIn the formula booklet	\(\frac{d}{dx}\Big(\frac uv\Big)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\)
Linear inside	\(\tfrac{d}{dx}e^{kx}=ke^{kx},\) \(\tfrac{d}{dx}\sin kx=k\cos kx,\) \(\tfrac{d}{dx}\ln(kx)=\tfrac1x\)
Tangent; normal gradient at \(x=a\)	\(y-f(a)=f'(a)(x-a);\) \(m_\text{normal}=-\frac{1}{f'(a)}\)
Increasing / decreasing	\(f'(x)>0;\) \(f'(x)<0\)
Stationary points	\(f'(x)=0:\) \(f''<0\text{ max},\) \(f''>0\text{ min}\)
Concave up \(f''>0\); point of inflexion: \(f''=0\) and \(f''\) changes sign
Integral of \(x^n\), \(n\ne-1\)In the formula booklet	\(\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\)
Standard integralsIn the formula booklet	\(\int\tfrac1x\,dx=\ln\|x\|+C,\) \(\int\sin x\,dx=-\cos x+C,\) \(\int\cos x\,dx=\sin x+C,\) \(\int e^x\,dx=e^x+C\)
Linear inside	\(\int f(ax+b)\,dx=\tfrac1aF(ax+b)+C\)
Reverse chain rule	\(\int f'\big(g(x)\big)g'(x)\,dx=f\big(g(x)\big)+C\)
Definite integral	\(\int_a^bf'(x)\,dx=f(b)-f(a)\)
Area, curve and \(x\)-axisIn the formula booklet	\(A=\int_a^b\|y\|\,dx\)

More formulas are on the full IB Maths AA SL formula sheet.

Worked example

Find the stationary points of y = x³ − 6x² + 9x + 1 and say what type each is.

dy/dx = 3x² − 12x + 9 = 3(x − 1)(x − 3) = 0, so x = 1 or x = 3
y(1) = 5 and y(3) = 1
d²y/dx² = 6x − 12: at x = 1 it is −6 < 0, at x = 3 it is 6 > 0

Answer: (1, 5) is a local maximum and (3, 1) is a local minimum

Common mistakes

Kinematics: speed vs velocity, change of direction, distance vs displacement
GDC in the wrong angle mode, and degrees in radian formulas
Forgetting to multiply by the original exponent when differentiating xⁿ
Treating dy/dx of a constant as anything other than 0

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

Estimate the gradient of a curve at a point using chords that get closer and closer, and describe this as a limit.
Find indefinite integrals of power functions and polynomials by reversing differentiation, always including the constant of integration.
Differentiate sin x, cos x and tan x and expressions built from them, such as 3 sin x − 2 cos x, working in radians.
Solve optimisation problems in which the function involves trigonometric, exponential or logarithmic terms, justifying that the value found is a maximum or minimum.
Integrate expressions such as 2x(x² + 1)⁴ and cos x · e^(sin x) by inspection or by a simple substitution, recognising a function times its derivative.
Integrate a(t) and v(t) with initial conditions to find velocity and displacement functions.

The printable sheet

Calculus knowledge organiser for IB Maths AA SL: one A4 page of key definitions, formulas, a worked example and common mistakes — Calculus knowledge organiser (IB Maths AA SL), A4. Download the PDF.

Revise it next

Other IB Maths AA SL topics: Number and algebra · Functions · Geometry and trigonometry · Statistics and probability · All IB Maths AA SL organisers