Knowledge organisers

Calculus: IB Maths AA HL 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

Integration by parts: ∫u dv = uv − ∫v du: use it for a product such as x eˣ or x sin x.
Implicit differentiation: Differentiate both sides with respect to x, multiplying by dy/dx each time you differentiate a term in y.
Maclaurin series: Writes f(x) as a power series using the derivatives of f at x = 0.
Separable differential equation: dy/dx = f(x)g(y) can be solved by writing ∫ dy/g(y) = ∫ f(x) 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.

First principlesIn the formula booklet	\(f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\)
Basic derivativesIn the formula booklet	\((x^n)'=nx^{n-1},\) \((\sin x)'=\cos x,\) \((\cos x)'=-\sin x,\) \((e^x)'=e^x,\) \((\ln x)'=\tfrac1x\)
Trig derivativesIn the formula booklet	\((\tan x)'=\sec^2x,\) \((\sec x)'=\sec x\tan x,\) \((\cosec x)'=-\cosec x\cot x,\) \((\cot x)'=-\cosec^2x\)
More derivativesIn the formula booklet	\((a^x)'=a^x\ln a,\) \((\log_ax)'=\tfrac1{x\ln a}\)
Inverse trigIn the formula booklet	\((\arcsin x)'=\tfrac1{\sqrt{1-x^2}},\) \((\arccos x)'=-\tfrac1{\sqrt{1-x^2}},\) \((\arctan x)'=\tfrac1{1+x^2}\)
Chain, product, quotientIn the formula booklet	\(\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx},\) \((uv)'=uv'+vu',\) \(\Big(\frac uv\Big)'=\frac{vu'-uv'}{v^2}\)
Implicit	\(\tfrac{d}{dx}\big(y^n\big)=ny^{n-1}\tfrac{dy}{dx},\) \(\tfrac{d}{dx}(xy)=x\tfrac{dy}{dx}+y\)
Related rates	\(\frac{dV}{dt}=\frac{dV}{dr}\cdot\frac{dr}{dt}\)
Tangent; normal	\(y-f(a)=f'(a)(x-a);\) \(m_\text{n}=-\tfrac1{f'(a)}\)
Max / min; inflexion	\(f'=0\text{ and }f''<0\ /\ f''>0;\) \(f''=0\text{ with sign change}\)
L'Hôpital, for \(\frac00\) or \(\frac\infty\infty\)	\(\lim\frac{f(x)}{g(x)}=\lim\frac{f'(x)}{g'(x)}\)
Standard integralsIn the formula booklet	\(\int x^n\,dx=\tfrac{x^{n+1}}{n+1}+C,\) \(\int\tfrac1x\,dx=\ln\|x\|+C,\) \(\int\sin x\,dx=-\cos x+C,\) \(\int\cos x\,dx=\sin x+C,\) \(\int e^x\,dx=e^x+C\)
More integralsIn the formula booklet	\(\int a^x\,dx=\tfrac{a^x}{\ln a}+C,\) \(\int\tfrac{dx}{a^2+x^2}=\tfrac1a\arctan\tfrac xa+C,\) \(\int\tfrac{dx}{\sqrt{a^2-x^2}}=\arcsin\tfrac xa+C\)
Learn	\(\int\sec^2x\,dx=\tan x+C,\) \(\int\tfrac{f'(x)}{f(x)}\,dx=\ln\|f(x)\|+C,\) \(\int f(ax+b)\,dx=\tfrac1aF(ax+b)+C\)
\(\sin^2x\), \(\cos^2x\)	\(\sin^2x=\tfrac12(1-\cos2x),\) \(\cos^2x=\tfrac12(1+\cos2x)\)
By partsIn the formula booklet	\(\int u\frac{dv}{dx}\,dx=uv-\int v\frac{du}{dx}\,dx\)

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

Worked example

Find ∫ x eˣ dx.

Let u = x and dv/dx = eˣ, so du/dx = 1 and v = eˣ
∫ x eˣ dx = x eˣ − ∫ eˣ dx

Answer: ∫ x eˣ dx = x eˣ − eˣ + C

Common mistakes

Arithmetic with limits, fractions and simultaneous equations on Paper 1
"Hence", "deduce" and "exact": using the wrong method
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.
Set up a function for a quantity to be maximised or minimised, find its stationary points and justify the optimum value in context.
Differentiate sec x, cosec x, cot x, aˣ and logₐ x and combine them with the chain, product and quotient rules.
Integrate sec²x, 1/(a² + x²) and 1/√(a² − x²), giving answers in terms of tan x, arctan and arcsin, and evaluate related definite integrals.
Choose between inspection, substitution, integration by parts and partial fractions for a given integral, and carry it through.
Evaluate indeterminate limits using L'Hôpital's rule more than once, or using Maclaurin series.

The printable sheet

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

Revise it next

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