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DP IB Maths: AA SL

Revision Notes

Home / IB / Maths: AA SL / DP / Revision Notes / 5. Calculus / 5.2 Further Differentiation / 5.2.6 Derivatives & Graphs


5.2.6 Derivatives & Graphs


Derivatives & Graphs

How are derivatives and graphs connected?

  • If the graph of a functionspace y equals f left parenthesis x right parenthesis is known, or can be sketched, then it is also possible to sketch the graphs of the derivativesspace y equals f apostrophe left parenthesis x right parenthesis andspace y equals f apostrophe apostrophe left parenthesis x right parenthesis
  • The key properties of a graph include
    • thebold space bold italic y-axis intercept
    • thebold space bold italic x-axis intercepts – the roots of the function; wherespace f left parenthesis x right parenthesis equals 0
    • stationary points; wherespace f apostrophe left parenthesis x right parenthesis equals 0
      • turning points – (local) minimum and maximum points
      • (horizontal) points of inflection
    • (non-stationary,space straight f apostrophe left parenthesis x right parenthesis not equal to 0) points of inflection
    • asymptotes – vertical and horizontal
    • intervals where the graph is increasing and decreasing
    • intervals where the graph is concave down and concave up
  • Not all graphs have all of these properties and not all can be determined without knowing the expression of the function
  • However questions will provide enough information to sketch
    • the shape of the graph
    • some of the key properties such as roots or turning points

How do I sketch the graph of y = f'(x) from the graph of y = f(x)?

  • The graph ofspace y equals f apostrophe left parenthesis x right parenthesis will have its
    • space x-axis intercepts at thespace x-coordinates of the stationary points ofspace y equals f left parenthesis x right parenthesis
    • turning points at thespace x-coordinates of the  points of inflection ofspace y equals f left parenthesis x right parenthesis
  • For intervals wherespace y equals f left parenthesis x right parenthesis is concave up,space y equals f apostrophe left parenthesis x right parenthesis will be increasing
  • For intervals wherespace y equals f left parenthesis x right parenthesis is concave down ,space y equals f apostrophe left parenthesis x right parenthesis will be decreasing
  • For intervals wherespace y equals f left parenthesis x right parenthesis is increasing,space y equals f apostrophe left parenthesis x right parenthesis will be positive
  • For intervals wherespace y equals f left parenthesis x right parenthesis is decreasing,space y equals f apostrophe left parenthesis x right parenthesis will be negative

How do I sketch the graph of y = f''(x) from the graph of y = f(x)?

  • First sketch the graph ofspace y equals f apostrophe left parenthesis x right parenthesis fromspace y equals f left parenthesis x right parenthesis, as per the above process
  • Then, using the same process, sketch the graph ofspace y equals f apostrophe apostrophe left parenthesis x right parenthesis from the graph ofspace y equals f apostrophe left parenthesis x right parenthesis
  • There are a couple of things you can deduce about the graph ofspace y equals f apostrophe apostrophe left parenthesis x right parenthesis directly from the graph ofspace y equals f left parenthesis x right parenthesis
    • The graph of space y equals f apostrophe apostrophe left parenthesis x right parenthesis will have its x-axis intercepts at the x-coordinates of the points of inflection of space y equals f left parenthesis x right parenthesis
    • For intervals wherespace y equals f left parenthesis x right parenthesis is concave up,space y equals f apostrophe apostrophe left parenthesis x right parenthesis will be positive
    • For intervals wherespace y equals f left parenthesis x right parenthesis is concave down,space y equals f apostrophe apostrophe left parenthesis x right parenthesis will be negative

5-2-6-ib-sl-aa-only-y-dy-d2y

Is it possible to sketch the graph of y = f(x) from the graph of a derivative?

  • It is possible to sketch a graph ofspace y equals f left parenthesis x right parenthesis by considering the reverse of the above
    • For intervals wherespace y equals f apostrophe left parenthesis x right parenthesis is positive,space y equals f left parenthesis x right parenthesis will be increasing but is not necessarily positive
    • For intervals wherespace y equals f apostrophe left parenthesis x right parenthesis is negative,space y equals f left parenthesis x right parenthesis will be decreasing but is not necessarily negative
    • Roots ofspace y equals f apostrophe left parenthesis x right parenthesis give thespace x-coordinates of the stationary points ofspace y equals f left parenthesis x right parenthesis
  • There are some properties of the graph ofspace y equals f left parenthesis x right parenthesis that cannot be determined from the graph ofspace y equals f apostrophe left parenthesis x right parenthesis
    • thebold space bold italic y-axis intercept
    • the intervals for whichspace y equals f left parenthesis x right parenthesis is positive and negative
    • the roots ofspace y equals f left parenthesis x right parenthesis
  • Unless a specific point the curve passes through is known, the constant of integration cannot be determined
    • the exact location of the curve will remain unknown
    • but it will still be possible to sketch its shape
  • If starting from the graph of the second derivative,space y equals f apostrophe apostrophe left parenthesis x right parenthesis, it is easier to sketch the graph ofspace y equals f apostrophe left parenthesis x right parenthesis first, then sketchspace y equals f left parenthesis x right parenthesis

Worked Example

The graph ofspace y equals f left parenthesis x right parenthesis is shown in the diagram below.

qwwD1Cx~_5-2-6-ib-sl-aa-only-we-quest

On separate diagrams sketch the graphs ofspace y equals f apostrophe left parenthesis x right parenthesis andspace y equals f apostrophe apostrophe left parenthesis x right parenthesis, labelling any roots and turning points.

5-2-6-ib-sl-aa-only-we-soltn



  • 1. Number & Algebra
    • 1.1 Number Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Laws of Indices
        • 1.2 Exponentials & Logs
          • 1.2.1 Introduction to Logarithms
            • 1.2.2 Laws of Logarithms
              • 1.2.3 Solving Exponential Equations
              • 1.3 Sequences & Series
                • 1.3.1 Language of Sequences & Series
                  • 1.3.2 Arithmetic Sequences & Series
                    • 1.3.3 Geometric Sequences & Series
                      • 1.3.4 Applications of Sequences & Series
                        • 1.3.5 Compound Interest & Depreciation
                        • 1.4 Proof & Reasoning
                          • 1.4.1 Proof
                          • 1.5 Binomial Theorem
                            • 1.5.1 Binomial Theorem
                          • 2. Functions
                            • 2.1 Linear Functions & Graphs
                              • 2.1.1 Equations of a Straight Line
                              • 2.2 Quadratic Functions & Graphs
                                • 2.2.1 Quadratic Functions
                                  • 2.2.2 Factorising & Completing the Square
                                    • 2.2.3 Solving Quadratics
                                      • 2.2.4 Quadratic Inequalities
                                        • 2.2.5 Discriminants
                                        • 2.3 Functions Toolkit
                                          • 2.3.1 Language of Functions
                                            • 2.3.2 Composite & Inverse Functions
                                              • 2.3.3 Graphing Functions
                                              • 2.4 Further Functions & Graphs
                                                • 2.4.1 Reciprocal & Rational Functions
                                                  • 2.4.2 Exponential & Logarithmic Functions
                                                    • 2.4.3 Solving Equations
                                                      • 2.4.4 Modelling with Functions
                                                      • 2.5 Transformations of Graphs
                                                        • 2.5.1 Translations of Graphs
                                                          • 2.5.2 Reflections of Graphs
                                                            • 2.5.3 Stretches of Graphs
                                                              • 2.5.4 Composite Transformations of Graphs
                                                            • 3. Geometry & Trigonometry
                                                              • 3.1 Geometry Toolkit
                                                                • 3.1.1 Coordinate Geometry
                                                                  • 3.1.2 Radian Measure
                                                                    • 3.1.3 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 Further Trigonometry
                                                                                • 3.4.1 The Unit Circle
                                                                                  • 3.4.2 Exact Values
                                                                                  • 3.5 Trigonometric Functions & Graphs
                                                                                    • 3.5.1 Graphs of Trigonometric Functions
                                                                                      • 3.5.2 Transformations of Trigonometric Functions
                                                                                        • 3.5.3 Modelling with Trigonometric Functions
                                                                                        • 3.6 Trigonometric Equations & Identities
                                                                                          • 3.6.1 Simple Identities
                                                                                            • 3.6.2 Double Angle Formulae
                                                                                              • 3.6.3 Relationship Between Trigonometric Ratios
                                                                                                • 3.6.4 Linear Trigonometric Equations
                                                                                                  • 3.6.5 Quadratic Trigonometric Equations
                                                                                                • 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 Tranformations 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 & 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
                                                                                                                                        • 4.6.3 Standardisation of Normal Variables
                                                                                                                                      • 5. Calculus
                                                                                                                                        • 5.1 Differentiation
                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                            • 5.2 Further Differentiation
                                                                                                                                              • 5.2.1 Differentiating Special Functions
                                                                                                                                                • 5.2.2 Techniques of Differentiation
                                                                                                                                                  • 5.2.3 Second Order Derivatives
                                                                                                                                                    • 5.2.4 Further Applications of Differentiation
                                                                                                                                                      • 5.2.5 Concavity & Points of Inflection
                                                                                                                                                        • 5.2.6 Derivatives & Graphs
                                                                                                                                                        • 5.3 Integration
                                                                                                                                                          • 5.3.1 Introduction to Integration
                                                                                                                                                            • 5.3.2 Applications of Integration
                                                                                                                                                            • 5.4 Further Integration
                                                                                                                                                              • 5.4.1 Integrating Special Functions
                                                                                                                                                                • 5.4.2 Techniques of Integration
                                                                                                                                                                  • 5.4.3 Definite Integrals
                                                                                                                                                                    • 5.4.4 Further Applications of Integration
                                                                                                                                                                    • 5.5 Optimisation
                                                                                                                                                                      • 5.5.1 Modelling with Differentiation
                                                                                                                                                                      • 5.6 Kinematics
                                                                                                                                                                        • 5.6.1 Kinematics Toolkit
                                                                                                                                                                          • 5.6.2 Calculus for Kinematics
                                                                                                                                                                        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.


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