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  1. Subjects
  3. Maths AA
  5. 5 Calculus

5 Calculus

An overview of Topic 5, Calculus: how differentiation and integration describe change and accumulation, how the subtopics build on each other, and how SL and HL differ.

  • IB Mathematics: Analysis and Approaches
  • SL and HL

19 subtopics in Calculus

Each one has its own page once something is published for it.

  1. 5.1Limits and the derivative24 questions · notes · 29 quick fire
  2. 5.2Increasing and decreasing functions36 questions · notes · 27 quick fire
  3. 5.3Differentiating polynomials70 questions · notes · 33 quick fire
  4. 5.4Tangents and normals74 questions · notes · 30 quick fire
  5. 5.5Introduction to integration37 questions · notes · 29 quick fire
  6. 5.6Chain, product and quotient rules131 questions · notes · 38 quick fire
  7. 5.7The second derivative57 questions · notes · 30 quick fire
  8. 5.8Stationary points, optimisation and inflexion166 questions · notes · 34 quick fire
  9. 5.9Kinematics41 questions · notes · 31 quick fire
  10. 5.10Indefinite integrals and substitution by inspection65 questions · notes · 41 quick fire
  11. 5.11Definite integrals and areas163 questions · notes · 39 quick fire
  12. 5.12Continuity, differentiability and first principlesHL only21 questions · notes · 39 quick fire
  13. 5.13Limits and L’Hôpital’s ruleHL only41 questions · notes · 38 quick fire
  14. 5.14Implicit differentiation and related ratesHL only26 questions · notes · 37 quick fire
  15. 5.15Derivatives of further functionsHL only51 questions · notes · 40 quick fire
  16. 5.16Integration by substitution and by partsHL only36 questions · notes · 39 quick fire
  17. 5.17Areas and volumes of revolutionHL only27 questions · notes · 38 quick fire
  18. 5.18First-order differential equationsHL only27 questions · notes · 40 quick fire
  19. 5.19Maclaurin seriesHL only28 questions · notes · 39 quick fire

An overview of 5 Calculus

5 min readLast reviewed 14 Oct 2026

Calculus is the mathematics of change and accumulation. Differential calculus measures how fast a quantity changes: the gradient of a curve at a point, the velocity of a moving object, the rate at which a population grows. Integral calculus does the reverse, adding up small changes to find a total: the area under a curve, the distance travelled from a velocity, the volume of a solid. The two are linked by the fundamental theorem of calculus, which shows that integration undoes differentiation.

Both halves rest on the idea of a limit. The gradient of a curve at a point is the limit of the gradients of chords, and an area under a curve is the limit of the sum of the areas of thinner and thinner rectangles. Calculus brings together almost every earlier topic: functions and their graphs (Topic 2), sequences and series (Topic 1), circular functions (Topic 3) and, at HL, continuous random variables (Topic 4).

YouTube
Watch how the area of a circle is found by slicing it into thin rings, an example of both central ideas of calculus: tiny changes, and adding up many of them.

How the subtopics build on each other

The topic starts with differentiation of simple functions and its uses, then introduces integration, and then extends both to a wider range of functions. Subtopics 5.12 to 5.19 are HL only.

Subtopic

Level

What it covers

5.1 Limits and the derivative

SL

Limits from tables and graphs; the derivative as a gradient and as a rate of change

5.2 Increasing and decreasing functions

SL

The sign of f′(x)f'(x)f′(x); sketching f′f'f′ from fff and the reverse

5.3 Differentiating polynomials

SL

The power rule for integer powers; gradients at points

5.4 Tangents and normals

SL

Equations of tangents and normals; tangents through an external point

5.5 Introduction to integration

SL

Anti-differentiation, the constant of integration, areas with a GDC

5.6 Chain, product and quotient rules

SL

Derivatives of xnx^nxn, sin⁡x\sin xsinx, cos⁡x\cos xcosx, exe^xex, ln⁡x\ln xlnx and their combinations

5.7 The second derivative

SL

f′′(x)f''(x)f′′(x), concavity and the graphs of fff, f′f'f′ and f′′f''f′′

5.8 Stationary points, optimisation and inflexion

SL

Maximum and minimum points, optimisation problems, points of inflexion

5.9 Kinematics

SL

Displacement, velocity, acceleration and distance travelled

5.10 Indefinite integrals and substitution by inspection

SL

Standard integrals, linear composites and the reverse chain rule

5.11 Definite integrals and areas

SL

Exact definite integrals, areas above and below the axis, areas between curves

5.12 Continuity, differentiability and first principles

HL

Continuity, differentiability, the derivative from first principles, higher derivatives

5.13 Limits and L'Hôpital's rule

HL

Indeterminate forms and L'Hôpital's rule

5.14 Implicit differentiation and related rates

HL

Implicit differentiation, related rates of change, harder optimisation

5.15 Derivatives of further functions

HL

tan⁡x\tan xtanx, sec⁡x\sec xsecx, axa^xax, log⁡ax\log_a xloga​x, inverse trigonometric functions and their integrals

5.16 Integration by substitution and by parts

HL

Substitution with changes of limits; integration by parts, including repeated use

5.17 Areas and volumes of revolution

HL

Areas with respect to yyy; volumes of revolution about either axis

5.18 First-order differential equations

HL

Slope fields, Euler's method, separable and homogeneous equations, integrating factors

5.19 Maclaurin series

HL

Power series for standard functions, and their use in approximations and limits

Subtopics 5.1 to 5.4 form the foundation: the meaning of a derivative, what its sign shows, how to find it for polynomials and how to use it for tangents. Subtopics 5.5, 5.10 and 5.11 build integration in three stages, while 5.6 to 5.9 extend differentiation to more functions and to applications. The HL subtopics then make the ideas rigorous (5.12, 5.13) and extend the methods to harder functions, differential equations and series.

Key concepts in calculus

  • Change. A derivative is a rate of change, and calculus describes any quantity that changes continuously: speed, growth, cooling.

  • Approximation. A tangent approximates a curve near a point, a sum of rectangles approximates an area, and a Maclaurin series approximates a function by a polynomial.

  • Modelling. Rates of change in context lead to optimisation problems, kinematics and, at HL, differential equations that model growth and decay.

  • Relationships and representation. The graphs of fff, f′f'f′ and f′′f''f′′ are three representations of one function, and features of one appear as features of the others.

  • Equivalence. Differentiation and integration are inverse processes, linked by the fundamental theorem of calculus.

Calculus at SL and at HL

SL students study subtopics 5.1 to 5.11, which cover differentiation of the standard functions with the chain, product and quotient rules, applications to graphs, optimisation and kinematics, and integration of standard functions, including areas. HL students study all of these and also 5.12 to 5.19. The HL material adds rigour (continuity, differentiability and first principles), a wider range of functions and integration techniques, volumes of revolution, differential equations and Maclaurin series.

Exam tip

Calculus is tested on every paper, at both levels, and carries a large share of the marks. On Paper 1 (no calculator), derivatives and integrals must be found by hand, with exact answers. On Paper 2, a GDC can find gradients, definite integrals, intersections and maximum points numerically. HL students also sit Paper 3, whose two extended problem-solving questions often use calculus to investigate a situation step by step, for example by building a model and analysing it with derivatives or differential equations.

From the formula booklet

The entries of the mathematics: analysis and approaches formula booklet that 5 uses. Each opens in the explained booklet.

  • Derivative of xnx^nxnTopic 5: Calculus › SL and HL › SL 5.3 Derivative of xnx^nxnf(x)=xn⇒f′(x)=nxn−1f(x) = x^n \Rightarrow f'(x) = nx^{n-1}f(x)=xn⇒f′(x)=nxn−1
  • Integral of xnx^nxnTopic 5: Calculus › SL and HL › SL 5.5 Integral of xnx^nxn∫xn dx=xn+1n+1+C\int x^n\,\mathrm{d}x = \frac{x^{n+1}}{n + 1} + C∫xndx=n+1xn+1​+C
  • Area between a curve y=f(x)y = f(x)y=f(x) and the xxx-axis, where f(x)>0f(x) > 0f(x)>0Topic 5: Calculus › SL and HL › SL 5.5 Area between a curve y=f(x)y = f(x)y=f(x) and the xxx-axis, where f(x)>0f(x) > 0f(x)>0A=∫aby dxA = \int_a^b y\,\mathrm{d}xA=∫ab​ydx
  • Derivative of sin⁡x\sin xsinxTopic 5: Calculus › SL and HL › SL 5.6 Derivative of sin⁡x\sin xsinxf(x)=sin⁡x⇒f′(x)=cos⁡xf(x) = \sin x \Rightarrow f'(x) = \cos xf(x)=sinx⇒f′(x)=cosx
  • Derivative of cos⁡x\cos xcosxTopic 5: Calculus › SL and HL › SL 5.6 Derivative of cos⁡x\cos xcosxf(x)=cos⁡x⇒f′(x)=−sin⁡xf(x) = \cos x \Rightarrow f'(x) = -\sin xf(x)=cosx⇒f′(x)=−sinx
  • Derivative of ex\mathrm{e}^xexTopic 5: Calculus › SL and HL › SL 5.6 Derivative of ex\mathrm{e}^xexf(x)=ex⇒f′(x)=exf(x) = \mathrm{e}^x \Rightarrow f'(x) = \mathrm{e}^xf(x)=ex⇒f′(x)=ex
  • Derivative of ln⁡x\ln xlnxTopic 5: Calculus › SL and HL › SL 5.6 Derivative of ln⁡x\ln xlnxf(x)=ln⁡x⇒f′(x)=1xf(x) = \ln x \Rightarrow f'(x) = \frac{1}{x}f(x)=lnx⇒f′(x)=x1​
  • Chain ruleTopic 5: Calculus › SL and HL › SL 5.6 Chain ruley=g(u), where u=f(x)⇒dydx=dydu×dudxy = g(u)\text{, where } u = f(x) \Rightarrow \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{\mathrm{d}y}{\mathrm{d}u} \times \frac{\mathrm{d}u}{\mathrm{d}x}y=g(u), where u=f(x)⇒dxdy​=dudy​×dxdu​
  • Product ruleTopic 5: Calculus › SL and HL › SL 5.6 Product ruley=uv⇒dydx=udvdx+vdudxy = uv \Rightarrow \frac{\mathrm{d}y}{\mathrm{d}x} = u\frac{\mathrm{d}v}{\mathrm{d}x} + v\frac{\mathrm{d}u}{\mathrm{d}x}y=uv⇒dxdy​=udxdv​+vdxdu​
  • Quotient ruleTopic 5: Calculus › SL and HL › SL 5.6 Quotient ruley=uv⇒dydx=vdudx−udvdxv2y = \frac{u}{v} \Rightarrow \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{v\frac{\mathrm{d}u}{\mathrm{d}x} - u\frac{\mathrm{d}v}{\mathrm{d}x}}{v^2}y=vu​⇒dxdy​=v2vdxdu​−udxdv​​

And 33 more in the formula booklet.

On this page

  • Subtopics
  • Notes
  • Formula booklet

Published for 5

Exam-style questions
547
Notes pages
20
Quick-fire questions
671
Key terms
90
Counted on 20 October 2026Open it in Augment

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