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DP IB Maths: AI HL

Revision Notes

Home / IB / Maths: AI HL / 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 functionspace 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 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 (velocity)
    • 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
  • The derivative is also called the gradient function

How are limits and derivatives linked?

  • Consider the point P on the graph of y equals 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 functionspace 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 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 f apostrophe left parenthesis x right parenthesis

  • Different variables may be used
    • e.g. If V equals f left parenthesis s right parenthesis then  fraction numerator straight d V over denominator straight d s end fraction equals 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 f left parenthesis x right parenthesis wherespace 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:
    • Ifspace f left parenthesis x right parenthesis equals x to the power of n thenspace f apostrophe left parenthesis x right parenthesis equals n x to the power of n minus 1 end exponent where n element of straight rational 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
    • Ifspace f left parenthesis x right parenthesis equals a x to the power of n thenspace 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 rational numbers and a is a constant
  • The alternative notation (tospace 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 1 half end exponent then fraction numerator straight d y over denominator straight d x end fraction equals negative 4 cross times 1 half cross times x to the power of 1 half minus 1 end exponent equals negative 2 x to the power of negative 1 half end exponent
  • Don't forget these two special cases:
    • Ifspace f left parenthesis x right parenthesis equals a x thenspace 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
    • Ifspace f left parenthesis x right parenthesis equals a thenspace 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
  • Funcitons involving roots will need to be rewritten as fractional powers of x first
    • e.g.  If space f left parenthesis x right parenthesis equals 2 square root of x then rewrite asspace f left parenthesis x right parenthesis equals 2 x to the power of 1 half end exponent and differentiate
  • Functions involving fractions with denominators in terms of x will need to be rewritten as negative powers of x first
    • e.g.  Ifspace f left parenthesis x right parenthesis equals 4 over x then rewrite asspace 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 rational powers so it is possible to differentiate any expression that is a sum or difference of powers of x
    • e.g.  Ifspace f left parenthesis x right parenthesis equals 5 x to the power of 4 minus 3 x to the power of 2 over 3 end exponent plus 4 then
      space f apostrophe left parenthesis x right parenthesis equals 5 cross times 4 x to the power of 4 minus 1 end exponent minus 3 cross times 2 over 3 x to the power of 2 over 3 minus 1 end exponent plus 0
      space f apostrophe left parenthesis x right parenthesis equals 20 x cubed minus 2 x to the power of negative 1 third end exponent
  • Products and quotients cannot be differentiated in this way so would need expanding/simplifying first
    • e.g.  Ifspace 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 tospace 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

What might I be asked to do once I’ve found the derivative (gradient function)?

  • The derivative can be used to find the gradient of a function at any point
    • The gradient of a function at a point is equal to the gradient of the tangent to the curve at that point
    • A question may refer to the gradient of the tangent
    • A GDC can be used to help find gradients or verify answers
  • Take extra care when differentiating negative and fractional powers of x
    • 1 half minus 1 equals negative 1 half but 1 third minus 1 equals negative 2 over 3

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 functionspace f left parenthesis x right parenthesis is given by

space f left parenthesis x right parenthesis equals 2 x cubed plus fraction numerator 4 over denominator square root of x end fraction

where x greater than 0.

a)
Find the derivative ofspace f left parenthesis x right parenthesis.
5-1-1-ib-sl-ai-aa-extra-we2-soltn-a

b)
Find the gradient of the tangent to the curve y equals f left parenthesis x right parenthesis at the point where x equals 4.

5-1-1-ib-sl-aa-vers-we2-soltn-a-b



  • 1. Number & Algebra
    • 1.1 Number Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Approximation & Estimation
          • 1.1.3 GDC: Solving Equations
          • 1.2 Exponentials & Logs
            • 1.2.1 Exponents
              • 1.2.2 Logarithms
              • 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
                      • 1.4 Financial Applications
                        • 1.4.1 Compound Interest & Depreciation
                          • 1.4.2 Amortisation & Annuities
                          • 1.5 Complex Numbers
                            • 1.5.1 Intro to Complex Numbers
                              • 1.5.2 Modulus & Argument
                                • 1.5.3 Introduction to Argand Diagrams
                                • 1.6 Further Complex Numbers
                                  • 1.6.1 Geometry of Complex Numbers
                                    • 1.6.2 Forms of Complex Numbers
                                      • 1.6.3 Applications of Complex Numbers
                                      • 1.7 Matrices
                                        • 1.7.1 Introduction to Matrices
                                          • 1.7.2 Operations with Matrices
                                            • 1.7.3 Determinants & Inverses
                                              • 1.7.4 Solving Systems of Linear Equations with Matrices
                                              • 1.8 Eigenvalues & Eigenvectors
                                                • 1.8.1 Eigenvalues & Eigenvectors
                                                  • 1.8.2 Applications of Matrices
                                                • 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 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
                                                                      • 2.4 Functions Toolkit
                                                                        • 2.4.1 Composite & Inverse 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
                                                                                • 2.6 Further Modelling with Functions
                                                                                  • 2.6.1 Properties of Further Graphs
                                                                                    • 2.6.2 Natural Logarithmic Models
                                                                                      • 2.6.3 Logistic Models
                                                                                        • 2.6.4 Piecewise Models
                                                                                      • 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 Simple Identities
                                                                                                              • 3.4.3 Solving Trigonometric Equations
                                                                                                              • 3.5 Voronoi Diagrams
                                                                                                                • 3.5.1 Voronoi Diagrams
                                                                                                                  • 3.5.2 Toxic Waste Dump Problem
                                                                                                                  • 3.6 Matrix Transformations
                                                                                                                    • 3.6.1 Matrix Transformations
                                                                                                                      • 3.6.2 Determinant of a Transformation Matrix
                                                                                                                      • 3.7 Vector Properties
                                                                                                                        • 3.7.1 Introduction to Vectors
                                                                                                                          • 3.7.2 Position & Displacement Vectors
                                                                                                                            • 3.7.3 Magnitude of a Vector
                                                                                                                              • 3.7.4 The Scalar Product
                                                                                                                                • 3.7.5 The Vector Product
                                                                                                                                  • 3.7.6 Components of Vectors
                                                                                                                                    • 3.7.7 Geometric Proof with Vectors
                                                                                                                                    • 3.8 Vector Equations of Lines
                                                                                                                                      • 3.8.1 Vector Equations of Lines
                                                                                                                                        • 3.8.2 Shortest Distances with Lines
                                                                                                                                        • 3.9 Modelling with Vectors
                                                                                                                                          • 3.9.1 Kinematics with Vectors
                                                                                                                                            • 3.9.2 Constant & Variable Velocity
                                                                                                                                            • 3.10 Graph Theory
                                                                                                                                              • 3.10.1 Introduction to Graph Theory
                                                                                                                                                • 3.10.2 Walks & Adjacency Matrices
                                                                                                                                                  • 3.10.3 Minimum Spanning Trees
                                                                                                                                                    • 3.10.4 Chinese Postman Problem
                                                                                                                                                      • 3.10.5 Travelling Salesman Problem
                                                                                                                                                        • 3.10.6 Bounds for Travelling Salesman Problem
                                                                                                                                                      • 4. Statistics & Probability
                                                                                                                                                        • 4.1 Statistics Toolkit
                                                                                                                                                          • 4.1.1 Sampling
                                                                                                                                                            • 4.1.2 Data Collection
                                                                                                                                                              • 4.1.3 Statistical Measures
                                                                                                                                                                • 4.1.4 Frequency Tables
                                                                                                                                                                  • 4.1.5 Linear Transformations of Data
                                                                                                                                                                    • 4.1.6 Outliers
                                                                                                                                                                      • 4.1.7 Univariate Data
                                                                                                                                                                        • 4.1.8 Interpreting Data
                                                                                                                                                                        • 4.2 Correlation & Regression
                                                                                                                                                                          • 4.2.1 Bivariate Data
                                                                                                                                                                            • 4.2.2 Correlation Coefficients
                                                                                                                                                                              • 4.2.3 Linear Regression
                                                                                                                                                                              • 4.3 Further Correlation & Regression
                                                                                                                                                                                • 4.3.1 Non-linear Regression
                                                                                                                                                                                  • 4.3.2 Logarithmic Scales
                                                                                                                                                                                    • 4.3.3 Linearising using Logarithms
                                                                                                                                                                                    • 4.4 Probability
                                                                                                                                                                                      • 4.4.1 Probability & Types of Events
                                                                                                                                                                                        • 4.4.2 Conditional Probability
                                                                                                                                                                                          • 4.4.3 Sample Space Diagrams
                                                                                                                                                                                          • 4.5 Probability Distributions
                                                                                                                                                                                            • 4.5.1 Discrete Probability Distributions
                                                                                                                                                                                              • 4.5.2 Expected Values
                                                                                                                                                                                              • 4.6 Random Variables
                                                                                                                                                                                                • 4.6.1 Linear Combinations of Random Variables
                                                                                                                                                                                                  • 4.6.2 Unbiased Estimates
                                                                                                                                                                                                  • 4.7 Binomial Distribution
                                                                                                                                                                                                    • 4.7.1 The Binomial Distribution
                                                                                                                                                                                                      • 4.7.2 Calculating Binomial Probabilities
                                                                                                                                                                                                      • 4.8 Normal Distribution
                                                                                                                                                                                                        • 4.8.1 The Normal Distribution
                                                                                                                                                                                                          • 4.8.2 Calculations with Normal Distribution
                                                                                                                                                                                                          • 4.9 Further Normal Distribution (inc Central Limit Theorem)
                                                                                                                                                                                                            • 4.9.1 Sample Mean Distribution
                                                                                                                                                                                                              • 4.9.2 Confidence Interval for the Mean
                                                                                                                                                                                                              • 4.10 Poisson Distribution
                                                                                                                                                                                                                • 4.10.1 Poisson Distribution
                                                                                                                                                                                                                  • 4.10.2 Calculating Poisson Probabilities
                                                                                                                                                                                                                  • 4.11 Hypothesis Testing
                                                                                                                                                                                                                    • 4.11.1 Hypothesis Testing
                                                                                                                                                                                                                      • 4.11.2 Chi-squared Test for Independence
                                                                                                                                                                                                                        • 4.11.3 Goodness of Fit Test
                                                                                                                                                                                                                        • 4.12 Further Hypothesis Testing
                                                                                                                                                                                                                          • 4.12.1 Hypothesis Testing for Mean (One Sample)
                                                                                                                                                                                                                            • 4.12.2 Hypothesis Testing for Mean (Two Sample)
                                                                                                                                                                                                                              • 4.12.3 Binomial Hypothesis Testing
                                                                                                                                                                                                                                • 4.12.4 Poisson Hypothesis Testing
                                                                                                                                                                                                                                  • 4.12.5 Hypothesis Testing for Correlation
                                                                                                                                                                                                                                    • 4.12.6 Type I & Type II Errors
                                                                                                                                                                                                                                    • 4.13 Transition Matrices & Markov Chains
                                                                                                                                                                                                                                      • 4.13.1 Markov Chains
                                                                                                                                                                                                                                        • 4.13.2 Transition Matrices
                                                                                                                                                                                                                                      • 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 Further Differentiation
                                                                                                                                                                                                                                                • 5.2.1 Differentiating Special Functions
                                                                                                                                                                                                                                                  • 5.2.2 Techniques of Differentiation
                                                                                                                                                                                                                                                    • 5.2.3 Related Rates of Change
                                                                                                                                                                                                                                                      • 5.2.4 Second Order Derivatives
                                                                                                                                                                                                                                                        • 5.2.5 Further Applications of Differentiation
                                                                                                                                                                                                                                                          • 5.2.6 Concavity & Points of Inflection
                                                                                                                                                                                                                                                          • 5.3 Integration
                                                                                                                                                                                                                                                            • 5.3.1 Trapezoid Rule: Numerical Integration
                                                                                                                                                                                                                                                              • 5.3.2 Introduction to Integration
                                                                                                                                                                                                                                                                • 5.3.3 Applications of Integration
                                                                                                                                                                                                                                                                • 5.4 Further Integration
                                                                                                                                                                                                                                                                  • 5.4.1 Integrating Special Functions
                                                                                                                                                                                                                                                                    • 5.4.2 Techniques of Integration
                                                                                                                                                                                                                                                                      • 5.4.3 Further Applications of Integration
                                                                                                                                                                                                                                                                        • 5.4.4 Volumes of Revolution
                                                                                                                                                                                                                                                                        • 5.5 Kinematics
                                                                                                                                                                                                                                                                          • 5.5.1 Kinematics Toolkit
                                                                                                                                                                                                                                                                            • 5.5.2 Calculus for Kinematics
                                                                                                                                                                                                                                                                            • 5.6 Differential Equations
                                                                                                                                                                                                                                                                              • 5.6.1 Modelling with Differential Equations
                                                                                                                                                                                                                                                                                • 5.6.2 Separation of Variables
                                                                                                                                                                                                                                                                                  • 5.6.3 Slope Fields
                                                                                                                                                                                                                                                                                    • 5.6.4 Approximate Solutions to Differential Equations
                                                                                                                                                                                                                                                                                    • 5.7 Further Differential Equations
                                                                                                                                                                                                                                                                                      • 5.7.1 Coupled Differential Equations
                                                                                                                                                                                                                                                                                        • 5.7.2 Second Order Differential Equations
                                                                                                                                                                                                                                                                                      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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