• IB
  • IB Docs (2) Team
    Logout
  • Maths
  • Biology
  • Chemistry
  • Physics
  • Combined Science
  • English Language
  • Geography
  • Other Subjects
GCSE Maths
Edexcel Topic QuestionsRevision NotesPast PapersPast Papers Questions
AQA Topic QuestionsRevision NotesPast Papers
OCR Topic QuestionsRevision NotesPast Papers
GCSE Biology
Edexcel Topic QuestionsRevision NotesPast Papers
AQA Topic QuestionsRevision NotesPast Papers
OCR Gateway Topic QuestionsRevision NotesPast Papers
GCSE Chemistry
Edexcel Topic QuestionsRevision NotesPast Papers
AQA Topic QuestionsRevision NotesPast Papers
OCR Gateway Topic QuestionsRevision NotesPast Papers
GCSE Physics
Edexcel Topic QuestionsRevision NotesPast Papers
AQA Topic QuestionsRevision NotesPast Papers
OCR Gateway Topic QuestionsRevision NotesPast Papers
GCSE Combined Science
Edexcel Combined: Biology Topic QuestionsRevision NotesPast Papers
Edexcel Combined: Chemistry Topic QuestionsRevision NotesPast Papers
Edexcel Combined: Physics Revision NotesPast Papers
AQA Combined: Biology Topic QuestionsRevision NotesPast Papers
AQA Combined: Chemistry Topic QuestionsRevision NotesPast Papers
AQA Combined: Physics Topic QuestionsRevision NotesPast Papers
OCR Gateway Combined: Biology Topic QuestionsRevision Notes
OCR Gateway Combined: Chemistry Revision Notes
OCR Gateway Combined: Physics Revision Notes
GCSE English Language
AQA Revision NotesPractice PapersPast Papers
Edexcel Past Papers
OCR Past Papers
GCSE Geography
AQA Topic QuestionsRevision Notes
Edexcel Topic Questions
GCSE Other Subjects
AQA English LiteratureBusinessComputer ScienceEconomicsFurther MathsGeographyHistoryPsychologySociologyStatistics
Edexcel English LiteratureBusinessComputer ScienceGeographyHistoryPsychologyStatistics
OCR English LiteratureBusinessComputer ScienceEconomicsPsychology
OCR Gateway GeographyHistory
  • Maths
  • Biology
  • Chemistry
  • Physics
  • Double Science
  • Economics
  • English Language
  • Geography
  • Other Subjects
IGCSE Maths
Edexcel Topic QuestionsRevision NotesPast PapersBronze-Silver-Gold Questions
CIE (Extended) Topic QuestionsRevision NotesPast Papers
CIE (Core) Topic QuestionsPast Papers
IGCSE Biology
Edexcel Topic QuestionsRevision NotesPast Papers
CIE 2020-2022 Topic QuestionsRevision NotesPast Papers
CIE 2023-2025 Topic QuestionsRevision NotesPast Papers
IGCSE Chemistry
Edexcel Topic QuestionsRevision NotesPast Papers
CIE 2020-2022 Topic QuestionsRevision NotesPast Papers
CIE 2023-2025 Topic QuestionsRevision NotesPast Papers
IGCSE Physics
Edexcel Topic QuestionsRevision NotesPast Papers
CIE 2020-2022 Topic QuestionsRevision NotesPast Papers
CIE 2023-2025 Topic QuestionsRevision NotesPast Papers
IGCSE Double Science
Edexcel Double: Biology Topic QuestionsRevision NotesPast Papers
Edexcel Double: Chemistry Topic QuestionsRevision NotesPast Papers
Edexcel Double: Physics Topic QuestionsRevision NotesPast Papers
IGCSE Economics
CIE Topic QuestionsRevision NotesPast Papers
IGCSE English Language
CIE Revision NotesPractice PapersPast Papers
Edexcel Past Papers
IGCSE Geography
CIE Revision NotesTopic QuestionsPast Papers
Edexcel Topic QuestionsRevision NotesPast Papers
IGCSE Other Subjects
CIE Additional MathsEnglish LiteratureBusinessComputer ScienceHistorySociology
Edexcel English LiteratureBusinessComputer ScienceHistoryFurther Maths
  • Maths
  • Biology
  • Chemistry
  • Physics
  • English Language
  • Other Subjects
AS Maths
Edexcel Pure MathsMechanicsStatistics
AQA Pure MathsMechanicsStatistics
OCR Pure MathsMechanicsStatistics
CIE Pure 1Pure 2MechanicsProbability & Statistics 1
Edexcel IAS Pure 1Pure 2MechanicsStatistics
AS Biology
AQA Topic QuestionsRevision NotesPast Papers
OCR Topic QuestionsRevision NotesPast Papers
CIE 2019-2021 Topic QuestionsRevision NotesPast Papers
CIE 2022-2024 Topic QuestionsRevision NotesPast Papers
Edexcel IAL Revision Notes
AS Chemistry
Edexcel Revision Notes
AQA Topic QuestionsRevision NotesPast Papers
OCR Revision Notes
CIE 2019-2021 Topic QuestionsRevision NotesPast Papers
CIE 2022-2024 Topic QuestionsRevision NotesPast Papers
Edexcel IAL Revision Notes
AS Physics
Edexcel Revision Notes
AQA Topic QuestionsRevision NotesPast Papers
OCR Revision NotesPast Papers
CIE 2019-2021 Topic QuestionsRevision NotesPast Papers
CIE 2022-2024 Topic QuestionsRevision NotesPast Papers
Edexcel IAL Revision Notes
AS English Language
AQA Past Papers
Edexcel Past Papers
OCR Past Papers
AS Other Subjects
AQA BusinessComputer ScienceEconomicsEnglish LiteratureFurther MathsGeographyHistoryPsychologySociology
Edexcel BusinessEconomicsEnglish LiteratureFurther MathsGeographyHistoryPsychology
OCR BusinessComputer ScienceEconomicsEnglish LiteratureFurther Maths AGeographyHistoryPsychologySociology
CIE Further Maths
  • Maths
  • Biology
  • Chemistry
  • Physics
  • English Language
  • Economics
  • Further Maths
  • Psychology
  • Other Subjects
A Level Maths
Edexcel Pure MathsMechanicsStatistics
AQA Pure MathsMechanicsStatistics
OCR Pure MathsMechanicsStatistics
CIE Pure 1Pure 3MechanicsProbability & Statistics 1Probability & Statistics 2
Edexcel IAL Pure 1Pure 2Pure 3Pure 4Mechanics 1Mechanics 2Statistics 1Statistics 2Decision 1
A Level Biology
Edexcel Topic QuestionsPast Papers
Edexcel A (SNAB) Revision Notes
AQA Topic QuestionsRevision NotesPast Papers
OCR Topic QuestionsRevision NotesPast PapersGold Questions
CIE 2019-2021 Topic QuestionsRevision NotesPast Papers
CIE 2022-2024 Topic QuestionsRevision NotesPast Papers
Edexcel IAL Topic QuestionsRevision NotesPast Papers
A Level Chemistry
Edexcel Topic QuestionsRevision NotesPast Papers
AQA Topic QuestionsRevision NotesPast Papers
OCR Topic QuestionsRevision NotesPast PapersGold Questions
CIE 2019-2021 Topic QuestionsRevision NotesPast Papers
CIE 2022-2024 Topic QuestionsRevision NotesPast Papers
Edexcel IAL Topic QuestionsRevision NotesPast Papers
A Level Physics
Edexcel Topic QuestionsRevision NotesPast Papers
AQA Topic QuestionsRevision NotesPast Papers
OCR Topic QuestionsRevision NotesPast Papers
CIE 2019-2021 Topic QuestionsRevision NotesPast Papers
CIE 2022-2024 Topic QuestionsRevision NotesPast Papers
Edexcel IAL Topic QuestionsRevision NotesPast Papers
A Level English Language
AQA Past Papers
CIE Past Papers
Edexcel Past Papers
OCR Past Papers
Edexcel IAL Past Papers
A Level Economics
Edexcel Topic QuestionsRevision NotesPast Papers
AQA Topic QuestionsPast Papers
OCR Past Papers
CIE Past Papers
A Level Further Maths
Edexcel Topic QuestionsRevision NotesPast Papers
AQA Past Papers
OCR Past Papers
CIE Past Papers
Edexcel IAL Past Papers
A Level Psychology
AQA Topic QuestionsRevision NotesPast Papers
CIE Past Papers
Edexcel Past Papers
OCR Past Papers
Edexcel IAL Past Papers
A Level Other Subjects
AQA BusinessComputer ScienceEconomicsEnglish LiteratureGeographyHistorySociology
CIE BusinessComputer ScienceEconomicsEnglish LiteratureGeographySociology
Edexcel BusinessEconomics AEnglish LiteratureGeographyHistory
OCR BusinessComputer ScienceEconomicsEnglish LiteratureGeographyHistorySociology
Edexcel IAL English LiteratureGeography
CIE IAL History
  • Biology
  • Chemistry
  • Physics
  • Other Subjects
O Level Biology
CIE Topic QuestionsPast Papers
O Level Chemistry
CIE Topic QuestionsPast Papers
O Level Physics
CIE Topic QuestionsPast Papers
O Level Other Subjects
CIE Additional MathsMaths D
  • Maths
  • Biology
  • Chemistry
  • Physics
Pre U Maths
CIE Topic QuestionsPast Papers
Pre U Biology
CIE Topic QuestionsPast Papers
Pre U Chemistry
CIE Topic QuestionsPast Papers
Pre U Physics
CIE Topic QuestionsPast Papers
  • Maths
  • Biology
  • Chemistry
  • Physics
  • Economics
IB Maths
Maths: AA HL Topic QuestionsRevision NotesPractice Papers
Maths: AI HL Topic QuestionsRevision NotesPractice Papers
Maths: AA SL Topic QuestionsRevision NotesPractice Papers
Maths: AI SL Topic QuestionsRevision NotesPractice Papers
IB Biology
Biology: SL Topic QuestionsRevision NotesPractice Papers
Biology: HL Topic QuestionsRevision NotesPractice Papers
IB Chemistry
Chemistry: SL Topic QuestionsRevision NotesPractice Papers
Chemistry: HL Topic QuestionsRevision NotesPractice Papers
IB Physics
Physics: SL Topic QuestionsRevision NotesPractice Papers
Physics: HL Topic QuestionsRevision NotesPractice Papers
IB Economics
Economics: SL Revision Notes

DP IB Maths: AI SL

Revision Notes

Home / IB / Maths: AI SL / DP / Revision Notes / 5. Calculus / 5.1 Differentiation / 5.1.1 Introduction to Differentiation


5.1.1 Introduction to Differentiation


Introduction to Derivatives

  • Before introducing a derivative, an understanding of a limit is helpful

What is a limit?

  • The limit of a function is the value a function (of x) approaches as x approaches a particular value from either side
    • Limits are of interest when the function is undefined at a particular value
    • For example, the function straight f left parenthesis x right parenthesis equals fraction numerator x to the power of 4 minus 1 over denominator x minus 1 end fraction will approach a limit as x approaches 1 from both below and above but is undefined at x equals 1 as this would involve dividing by zero

What might I be asked about limits?

  • You may be asked to predict or estimate limits from a table of function values or from the graph of y equals straight f left parenthesis x right parenthesis
  • You may be asked to use your GDC to plot the graph and use values from it to estimate a limit

What is a derivative?

  • Calculus is about rates of change
    • the way a car’s position on a road changes is its speed
    • the way the car’s speed changes is its acceleration
  • The gradient (rate of change) of a (non-linear) function varies with x
  • The derivative of a function is a function that relates the gradient to the value of x
  • It is also called the gradient function

How are limits and derivatives linked?

  • Consider the point P on the graph of y equals straight f left parenthesis x right parenthesis as shown below
    • left square bracket P Q subscript i right square bracket is a series of chords

5-1-2-definiton-of-derivatives-diagram-1

  • The gradient of the function straight f left parenthesis x right parenthesis at the point P is equal to the gradient of the tangent at point P
  • The gradient of the tangent at point P is the limit of the gradient of the chords left square bracket P Q subscript i right square bracket as point Q ‘slides’ down the curve and gets ever closer to point P
  • The gradient of the function changes as x changes
  • The derivative is the function that calculates the gradient from the value x

What is the notation for derivatives?

  • For the function y equals straight f left parenthesis x right parenthesis the derivative, with respect to x, would be written as

fraction numerator straight d y over denominator straight d x end fraction equals straight f apostrophe left parenthesis x right parenthesis

  • Different variables may be used
    • e.g. If V equals straight f left parenthesis s right parenthesis then  fraction numerator straight d V over denominator straight d s end fraction equals straight f apostrophe left parenthesis s right parenthesis

What might I be asked about derivatives?

  • You may be asked to use the graphing features of your GDC to find the gradients of a function at different values of x
  • From a series of gradient values, you may be asked to suggest an expression for the derivative (gradient function) of a function

Worked Example

The graph of y equals straight f left parenthesis x right parenthesis where straight f left parenthesis x right parenthesis equals x cubed minus 2 passes through the points P left parenthesis 2 comma space 6 right parenthesis comma space A left parenthesis 2.3 comma space 10.167 right parenthesis comma space B left parenthesis 2.1 comma space 7.261 right parenthesis and C left parenthesis 2.05 comma space 6.615125 right parenthesis.

a)
Find the gradient of the chords left square bracket P A right square bracket comma space left square bracket P B right square bracket and left square bracket P C right square bracket.

5-1-1-ib-sl-ai-aa-we1-soltn-a

b)
Estimate the gradient of the tangent to the curve at the point P.

5-1-1-ib-sl-ai-aa-we1-soltn-b

c)
Use your GDC to find the gradient of the tangent at the pont P.

5-1-1-ib-sl-ai-aa-we1-soltn-c

Differentiating Powers of x

What is differentiation?

  • Differentiation is the process of finding an expression of the derivative (gradient function) from the expression of a function

How do I differentiate powers of x?

  • Powers of x are differentiated according to the following formula:
    • If straight f left parenthesis x right parenthesis equals x to the power of n then straight f apostrophe left parenthesis x right parenthesis equals n x to the power of n minus 1 end exponent where n element of straight integer numbers
    • This is given in the formula booklet
  • If the power of x is multiplied by a constant then the derivative is also multiplied by that constant
    • If straight f left parenthesis x right parenthesis equals a x to the power of n then straight f apostrophe left parenthesis x right parenthesis equals a n x to the power of n minus 1 end exponent where n element of straight integer numbers and a is a constant
  • The alternative notation (to straight f apostrophe left parenthesis x right parenthesis) is to use fraction numerator straight d y over denominator straight d x end fraction
    • If y equals a x to the power of n then fraction numerator straight d y over denominator straight d x end fraction equals a n x to the power of n minus 1 end exponent
      • e.g.  If y equals negative 4 x to the power of 5 then fraction numerator straight d y over denominator straight d x end fraction equals negative 4 cross times 5 x to the power of 5 minus 1 end exponent equals negative 20 x to the power of 4
  • Don't forget these two special cases:
    • If straight f left parenthesis x right parenthesis equals a x then straight f apostrophe left parenthesis x right parenthesis equals a
      • e.g.  If y equals 6 x then fraction numerator straight d y over denominator straight d x end fraction equals 6
    • If straight f left parenthesis x right parenthesis equals a then straight f apostrophe left parenthesis x right parenthesis equals 0
      • e.g.  If y equals 5 then fraction numerator straight d y over denominator straight d x end fraction equals 0
    • These allow you to differentiate linear terms in x and constants
  • Functions involving fractions with denominators in terms of x will need to be rewritten as negative powers of x first
    • If straight f left parenthesis x right parenthesis equals 4 over x then rewrite as straight f left parenthesis x right parenthesis equals 4 x to the power of negative 1 end exponent and differentiate

How do I differentiate sums and differences of powers of x?

  •  The formulae for differentiating powers of x apply to all integer powers so it is possible to differentiate any expression that is a sum or difference of powers of x
    • e.g.  If straight f left parenthesis x right parenthesis equals 5 x to the power of 4 plus 2 x cubed minus 3 x plus 4 then
      straight f apostrophe left parenthesis x right parenthesis equals 5 cross times 4 x to the power of 4 minus 1 end exponent plus 2 cross times 3 x to the power of 3 minus 1 end exponent minus 3 plus 0
      straight f apostrophe left parenthesis x right parenthesis equals 20 x cubed plus 6 x squared minus 3
  • Products and quotients cannot be differentiated in this way so would need expanding/simplifying first
    • e.g.  If straight f left parenthesis x right parenthesis equals left parenthesis 2 x minus 3 right parenthesis left parenthesis x squared minus 4 right parenthesis then expand to straight f left parenthesis x right parenthesis equals 2 x cubed minus 3 x squared minus 8 x plus 12 which is a sum/difference of powers of x and can be differentiated

Exam Tip

  • A common mistake is not simplifying expressions before differentiating
    • The derivative of open parentheses x squared plus 3 close parentheses open parentheses x cubed minus 2 x plus 1 close parentheses can not be found by multiplying the derivatives of open parentheses x squared plus 3 close parentheses and open parentheses x cubed minus 2 x plus 1 close parentheses

Worked Example

The function straight f left parenthesis x right parenthesis is given by

straight f left parenthesis x right parenthesis equals x cubed minus 2 x squared plus 3 minus 4 over x cubed

a)
Find the derivative of straight f left parenthesis x right parenthesis.
5-1-1-ib-sl-ai-aa-we2-soltn-a
b)
Find the gradient of the tangent to the curve y equals straight f left parenthesis x right parenthesis at the points where x equals negative 1 and x equals 1.

5-1-1-ib-sl-ai-aa-we2-soltn-b



  • 1. Number & Algebra
    • 1.1 Number Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Exponents & Logarithms
          • 1.1.3 Approximation & Estimation
            • 1.1.4 GDC: Solving Equations
            • 1.2 Sequences & Series
              • 1.2.1 Language of Sequences & Series
                • 1.2.2 Arithmetic Sequences & Series
                  • 1.2.3 Geometric Sequences & Series
                    • 1.2.4 Applications of Sequences & Series
                    • 1.3 Financial Applications
                      • 1.3.1 Compound Interest & Depreciation
                        • 1.3.2 Amortisation & Annuities
                      • 2. Functions
                        • 2.1 Linear Functions & Graphs
                          • 2.1.1 Equations of a Straight Line
                          • 2.2 Further Functions & Graphs
                            • 2.2.1 Functions
                              • 2.2.2 Graphing Functions
                                • 2.2.3 Properties of Graphs
                                • 2.3 Modelling with Functions
                                  • 2.3.1 Linear & Piecewise Models
                                    • 2.3.2 Quadratic & Cubic Models
                                      • 2.3.3 Exponential Models
                                        • 2.3.4 Direct & Inverse Variation
                                          • 2.3.5 Sinusoidal Models
                                            • 2.3.6 Strategy for Modelling Functions
                                          • 3. Geometry & Trigonometry
                                            • 3.1 Geometry Toolkit
                                              • 3.1.1 Coordinate Geometry
                                                • 3.1.2 Arcs & Sectors
                                                • 3.2 Geometry of 3D Shapes
                                                  • 3.2.1 3D Coordinate Geometry
                                                    • 3.2.2 Volume & Surface Area
                                                    • 3.3 Trigonometry
                                                      • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                        • 3.3.2 Non Right-Angled Trigonometry
                                                          • 3.3.3 Applications of Trigonometry & Pythagoras
                                                          • 3.4 Voronoi Diagrams
                                                            • 3.4.1 Voronoi Diagrams
                                                              • 3.4.2 Toxic Waste Dump Problem
                                                            • 4. Statistics & Probability
                                                              • 4.1 Statistics Toolkit
                                                                • 4.1.1 Sampling & Data Collection
                                                                  • 4.1.2 Statistical Measures
                                                                    • 4.1.3 Frequency Tables
                                                                      • 4.1.4 Linear Transformations of Data
                                                                        • 4.1.5 Outliers
                                                                          • 4.1.6 Univariate Data
                                                                            • 4.1.7 Interpreting Data
                                                                            • 4.2 Correlation & Regression
                                                                              • 4.2.1 Bivariate data
                                                                                • 4.2.2 Correlation Coefficients
                                                                                  • 4.2.3 Linear Regression
                                                                                  • 4.3 Probability
                                                                                    • 4.3.1 Probability & Types of Events
                                                                                      • 4.3.2 Conditional Probability
                                                                                        • 4.3.3 Sample Space Diagrams
                                                                                        • 4.4 Probability Distributions
                                                                                          • 4.4.1 Discrete Probability Distributions
                                                                                            • 4.4.2 Expected Values
                                                                                            • 4.5 Binomial Distribution
                                                                                              • 4.5.1 The Binomial Distribution
                                                                                                • 4.5.2 Calculating Binomial Probabilities
                                                                                                • 4.6 Normal Distribution
                                                                                                  • 4.6.1 The Normal Distribution
                                                                                                    • 4.6.2 Calculations with Normal Distribution
                                                                                                    • 4.7 Hypothesis Testing
                                                                                                      • 4.7.1 Hypothesis Testing
                                                                                                        • 4.7.2 Chi-squared Test for Independence
                                                                                                          • 4.7.3 Goodness of Fit Test
                                                                                                            • 4.7.4 The t-test
                                                                                                          • 5. Calculus
                                                                                                            • 5.1 Differentiation
                                                                                                              • 5.1.1 Introduction to Differentiation
                                                                                                                • 5.1.2 Applications of Differentiation
                                                                                                                  • 5.1.3 Modelling with Differentiation
                                                                                                                  • 5.2 Integration
                                                                                                                    • 5.2.1 Trapezoid Rule: Numerical Integration
                                                                                                                      • 5.2.2 Introduction to Integration
                                                                                                                        • 5.2.3 Applications of Integration
                                                                                                                      Paul Freeman

                                                                                                                      Author: Paul

                                                                                                                      Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.


                                                                                                                      Save My Exams Logo
                                                                                                                      Resources
                                                                                                                      Home Join Support

                                                                                                                      Members
                                                                                                                      Members Home Account Logout

                                                                                                                      Company
                                                                                                                      About Us Contact Us Jobs Terms Privacy Facebook Twitter

                                                                                                                      Quick Links
                                                                                                                      GCSE Revision Notes IGCSE Revision Notes A Level Revision Notes Biology Chemistry Physics Maths 2022 Advance Information

                                                                                                                       
                                                                                                                      © IB Documents (2) Team & u/aimlesskr
                                                                                                                      IBO was not involved in the production of, and does not endorse, the resources created by Save My Exams.