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DP IB Maths: AA HL

Revision Notes

Home / IB / Maths: AA HL / DP / Revision Notes / 5. Calculus / 5.8 Advanced Differentiation / 5.8.4 Differentiating Further Functions


5.8.4 Differentiating Further Functions


This Revision Note focuses on the results and derivations of results involving the less common trigonometric, exponential and logarithmic functions.  As with any function, questions may go on to ask about gradients, tangents, normals and stationary points.

Differentiating Reciprocal Trigonometric Functions

What are the reciprocal trigonometric functions?

  • Secant, cosecant and cotangent and abbreviated and defined as

                          sec space x equals fraction numerator 1 over denominator cos space x end fraction          cosec space x equals fraction numerator 1 over denominator sin space x end fraction          cot space x equals fraction numerator 1 over denominator tan space x end fraction

  • Remember that for calculus, angles need to be measured in radians
    • theta may be used instead of x
  • cosec space x is sometimes further abbreviated to csc space x

What are the derivatives of the reciprocal trigonometric functions?

  • f left parenthesis x right parenthesis equals sec space x
    • f apostrophe left parenthesis x right parenthesis equals sec space x space tan space x
  • f left parenthesis x right parenthesis equals cosec space x
    • f apostrophe left parenthesis x right parenthesis equals negative cosec space x space cot space x
  • f left parenthesis x right parenthesis equals cot space x
    • f apostrophe left parenthesis x right parenthesis equals negative cosec squared space x
  • These are given in the formula booklet

How do I show or prove the derivatives of the reciprocal trigonometric functions?

  • For y equals sec space x
    • Rewrite, y equals fraction numerator 1 over denominator cos space x end fraction
    • Use quotient rule, fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator cos space x left parenthesis 0 right parenthesis minus left parenthesis 1 right parenthesis left parenthesis negative sin space x right parenthesis over denominator cos squared space x end fraction
    • Rearrange, fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator sin space x over denominator cos squared space x end fraction
    • Separate, fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator 1 over denominator cos space x end fraction cross times fraction numerator sin space x over denominator cos space x end fraction
    • Rewrite, fraction numerator straight d y over denominator straight d x end fraction equals sec space x space tan space x
  • Similarly, for y equals cosec space x
    • y equals fraction numerator 1 over denominator sin space x end fraction
    • fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator sin space x left parenthesis 0 right parenthesis minus left parenthesis 1 right parenthesis cos space x over denominator sin squared space x end fraction
    • fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator negative cos space x over denominator sin squared space x end fraction
    • fraction numerator straight d y over denominator straight d x end fraction equals negative fraction numerator 1 over denominator sin space x end fraction cross times fraction numerator cos space x over denominator sin space x end fraction
    • fraction numerator straight d y over denominator straight d x end fraction equals negative cosec space x space cot space x

What do the derivatives of reciprocal trig look like with a linear functions of x?

  • For linear functions of the form ax+b
    • f left parenthesis x right parenthesis equals sec open parentheses a x plus b close parentheses
      • f apostrophe left parenthesis x right parenthesis equals a space sec space open parentheses a x plus b close parentheses space tan space open parentheses a x plus b close parentheses
    • f left parenthesis x right parenthesis equals cosec space left parenthesis a x plus b right parenthesis
      • f apostrophe left parenthesis x right parenthesis equals negative a space cosec space left parenthesis a x plus b right parenthesis space cot space open parentheses a x plus b close parentheses
    • f left parenthesis x right parenthesis equals cot space left parenthesis a x plus b right parenthesis
      • f apostrophe left parenthesis x right parenthesis equals negative a space cosec squared space left parenthesis a x plus b right parenthesis
    • These are not given in the formula booklet
      • they can be derived from chain rule
      • they are not essential to remember

Exam Tip

  • Even if you think you have remembered these derivatives, always use the formula booklet to double check
    • those squares and negatives are easy to get muddled up!
  • Where two trig functions are involved in the derivative be careful with the angle multiple;  x comma space 2 x comma space 3 x, etc
    • An example of a common mistake is differentiating y equals c o s e c space 3 x
      • fraction numerator d y over denominator d x end fraction equals negative 3 italic space c o s e c space x space c o t space 3 x  instead of   fraction numerator d y over denominator d x end fraction equals negative 3 space c o s e c space 3 x space c o t space 3 x

Worked Example

Curve C has equation y equals 2 cot open parentheses 3 x minus pi over 8 close parentheses.

a)
Show that the derivative of cot space x is negative cosec squared space x.

5-8-3-ib-hl-aa-only-we1a-soltn

b)       Find fraction numerator straight d y over denominator straight d x end fraction for curve C.

5-8-3-ib-hl-aa-only-we1b-soltn

c)       Find the gradient of curve C at the point where x equals fraction numerator 7 pi over denominator 24 end fraction.

5-8-3-ib-hl-aa-only-we1c-soltn

Differentiating Inverse Trigonometric Functions

What are the inverse trigonometric functions?

  • arcsin, arccos and arctan are functions defined as the inverse functions of sine, cosine and tangent respectively
    •  arcsin open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses equals straight pi over 3 which is equivalent to sin space open parentheses pi over 3 close parentheses equals fraction numerator square root of 3 over denominator 2 end fraction
    •  arctan left parenthesis negative 1 right parenthesis equals fraction numerator 3 pi over denominator 4 end fraction which is equivalent to tan open parentheses fraction numerator 3 pi over denominator 4 end fraction close parentheses equals negative 1

What are the derivatives of the inverse trigonometric functions?

  • f left parenthesis x right parenthesis equals arcsin space x
    • f apostrophe left parenthesis x right parenthesis equals fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction
  • f left parenthesis x right parenthesis equals arccos space x
    • f apostrophe left parenthesis x right parenthesis equals negative fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction
  • f left parenthesis x right parenthesis equals arctan space x
    • f apostrophe left parenthesis x right parenthesis equals fraction numerator 1 over denominator 1 plus x squared end fraction
  • Unlike other derivatives these look completely unrelated at first
    • their derivation involves use of the identity cos squared space x plus sin squared space x identical to 1
    • hence the squares and square roots!
  • All three are given in the formula booklet
  • Note with the derivative of arctan space x that open parentheses 1 plus x squared close parentheses is the same as open parentheses x squared plus 1 close parentheses

How do I show or prove the derivatives of the inverse trigonometric functions?

  • For y equals arcsin space x
    • Rewrite, sin space y equals x
    • Differentiate implicitly, cos space y fraction numerator straight d y over denominator straight d x end fraction equals 1
    • Rearrange, fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator 1 over denominator cos space y end fraction
    • Using the identity cos squared space y identical to 1 minus sin squared space y rewrite, fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator 1 over denominator square root of 1 minus sin squared space y end root end fraction
    • Since, sin space y equals x, fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction
  • Similarly, for y equals arccos space x
    • cos space y equals x
    • negative sin space y fraction numerator straight d y over denominator straight d x end fraction equals 1
    • fraction numerator straight d y over denominator straight d x end fraction equals negative fraction numerator 1 over denominator sin space y end fraction
    • fraction numerator straight d y over denominator straight d x end fraction equals negative fraction numerator 1 over denominator square root of 1 minus cos squared space y end root end fraction
    • fraction numerator straight d y over denominator straight d x end fraction equals negative fraction numerator 1 over denominator square root of 1 minus x squared end root end fraction
  • Notice how the derivative of y equals arcsin space x is positive but is negative for y equals arccos space x
    • This subtle but crucial difference can be seen in their graphs
      • y equals arcsin space x has a positive gradient for all values of x in its domain
      • y equals arccos space x has a negative gradient for all values of x in its domain

What do the derivative of inverse trig look like with a linear function of x?

  • For linear functions of the form a x plus b
  • f left parenthesis x right parenthesis equals arcsin open parentheses a x plus b close parentheses
    • f apostrophe left parenthesis x right parenthesis equals fraction numerator a over denominator square root of 1 minus left parenthesis a x plus b right parenthesis squared end root end fraction
  • f left parenthesis x right parenthesis equals arccos open parentheses a x plus b close parentheses
    • f apostrophe left parenthesis x right parenthesis equals fraction numerator a over denominator square root of 1 minus left parenthesis a x plus b right parenthesis squared end root end fraction
  • f left parenthesis x right parenthesis equals arctan open parentheses a x plus b close parentheses
    • f apostrophe left parenthesis x right parenthesis equals fraction numerator a over denominator 1 plus left parenthesis a x plus b right parenthesis squared end fraction
  • These are not in the formula booklet 
    • they can be derived from chain rule
    • they are not essential to remember
    • they are not commonly used

Exam Tip

  • For space f left parenthesis x right parenthesis equals arctan space x the terms on the denominator can be reversed (as they are being added rather than subtracted)
    • space f apostrophe left parenthesis x right parenthesis equals fraction numerator 1 over denominator 1 plus x squared end fraction equals fraction numerator 1 over denominator x squared plus 1 end fraction
    • Don't be fooled by this, it sounds obvious but on awkward "show that" questions it can be off-putting!

Worked Example

a)       Show that the derivative of arctan space x is fraction numerator 1 over denominator 1 plus x squared end fraction

5-8-3-ib-hl-aa-only-we2a-soltn

b)
Find the derivative of arctan left parenthesis 5 x cubed minus 2 x right parenthesis.

5-8-3-ib-hl-aa-only-we2b-soltn

Differentiating Exponential & Logarithmic Functions

What are exponential and logarithmic functions?

  • Exponential functions have term(s) where the variable (x) is the power (exponent)
    • In general, these would be of the form y equals a to the power of x
      • The special case of this is when a equals e, i.e.  y equals e to the power of x
  • Logarithmic functions have term(s) where the logarithms of the variable (x) are involved
    • In general, these would be of the form y equals log subscript a x
      • The special case of this is when a equals e, i.e.  y equals log subscript e x equals ln space x

What are the derivatives of exponential functions?

  • The first two results, of the special cases above, have been met before
    • f left parenthesis x right parenthesis equals e to the power of x comma space space f apostrophe left parenthesis x right parenthesis equals e to the power of x
    • f left parenthesis x right parenthesis equals ln space x comma space space f apostrophe left parenthesis x right parenthesis equals 1 over x
    • These are given in the formula booklet
  • For the general forms of exponentials and logarithms
    • f left parenthesis x right parenthesis equals a to the power of x
      • f apostrophe left parenthesis x right parenthesis equals a to the power of x left parenthesis ln space a right parenthesis
    • f left parenthesis x right parenthesis equals log subscript a x
      • f apostrophe left parenthesis x right parenthesis equals fraction numerator 1 over denominator x ln space a end fraction
    • These are also given in the formula booklet

How do I show or prove the derivatives of exponential and logarithmic functions?

  • For y equals a to the power of x
    • Take natural logarithms of both sides, ln space y equals x ln space a
    • Use the laws of logarithms, ln space y equals x ln space a
    • Differentiate, implicitly, 1 over y fraction numerator straight d y over denominator straight d x end fraction equals ln space a
    • Rearrange, fraction numerator straight d y over denominator straight d x end fraction equals y ln space a
    • Substitute for y, fraction numerator straight d y over denominator straight d x end fraction equals a to the power of x ln space a
  • For y equals log subscript a x
    • Rewrite, x equals a to the power of y
    • Differentiate x with respect to y, using the above result, fraction numerator straight d x over denominator straight d y end fraction equals a to the power of y ln space a
    • Using fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator 1 over denominator begin display style fraction numerator straight d x over denominator straight d y end fraction end style end fraction, fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator 1 over denominator a to the power of y ln space a end fraction
    • Substitute for y, fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator 1 over denominator a to the power of log subscript a x end exponent ln space a end fraction
    • Simplify, fraction numerator straight d y over denominator straight d x end fraction equals fraction numerator 1 over denominator x ln space a end fraction

What do the derivatives of exponentials and logarithms look like with a linear functions of x?

  • For linear functions of the form p x plus q
    • f left parenthesis x right parenthesis equals a to the power of p x plus q end exponent
      • f apostrophe left parenthesis x right parenthesis equals p a to the power of p x plus q end exponent open parentheses ln space a close parentheses
    • f left parenthesis x right parenthesis equals log subscript a left parenthesis p x plus q right parenthesis
      • f apostrophe left parenthesis x right parenthesis equals fraction numerator p over denominator open parentheses p x plus q close parentheses ln space a end fraction
    • These are not in the formula booklet
      • they can be derived from chain rule
      • they are not essential to remember

Exam Tip

  • For questions that require the derivative in a particular format, you may need to use the laws of logarithms
    • With ln appearing in denominators be careful with the division law
      • ln space stretchy left parenthesis a over b stretchy right parenthesis equals ln space a space minus space ln space b
      • but  fraction numerator ln space a over denominator ln space b end fraction  cannot be simplified (unless there is some numerical connection between a and b)

Worked Example

a)
Find the derivative of a to the power of 3 x minus 2 end exponent.

5-8-3-ib-hl-aa-only-we3a-soltn

b)       Find an expression for fraction numerator straight d y over denominator straight d x end fraction given that y equals log subscript 5 open parentheses 2 x cubed close parentheses

5-8-3-ib-hl-aa-only-we3b-soltn



  • 1. Number & Algebra
    • 1.1 Number & Algebra Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Laws of Indices
          • 1.1.3 Partial Fractions
          • 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 Simple Proof & Reasoning
                            • 1.4.1 Proof
                            • 1.5 Further Proof & Reasoning
                              • 1.5.1 Proof by Induction
                                • 1.5.2 Proof by Contradiction
                                • 1.6 Binomial Theorem
                                  • 1.6.1 Binomial Theorem
                                    • 1.6.2 Extension of The Binomial Theorem
                                    • 1.7 Permutations & Combinations
                                      • 1.7.1 Counting Principles
                                        • 1.7.2 Permutations & Combinations
                                        • 1.8 Complex Numbers
                                          • 1.8.1 Intro to Complex Numbers
                                            • 1.8.2 Modulus & Argument
                                              • 1.8.3 Introduction to Argand Diagrams
                                              • 1.9 Further Complex Numbers
                                                • 1.9.1 Geometry of Complex Numbers
                                                  • 1.9.2 Forms of Complex Numbers
                                                    • 1.9.3 Complex Roots of Polynomials
                                                      • 1.9.4 De Moivre's Theorem
                                                        • 1.9.5 Roots of Complex Numbers
                                                        • 1.10 Systems of Linear Equations
                                                          • 1.10.1 Systems of Linear Equations
                                                            • 1.10.2 Algebraic Solutions
                                                          • 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 Symmetry of Functions
                                                                                • 2.3.4 Graphing Functions
                                                                                • 2.4 Other Functions & Graphs
                                                                                  • 2.4.1 Exponential & Logarithmic Functions
                                                                                    • 2.4.2 Solving Equations
                                                                                      • 2.4.3 Modelling with Functions
                                                                                      • 2.5 Reciprocal & Rational Functions
                                                                                        • 2.5.1 Reciprocal & Rational Functions
                                                                                        • 2.6 Transformations of Graphs
                                                                                          • 2.6.1 Translations of Graphs
                                                                                            • 2.6.2 Reflections of Graphs
                                                                                              • 2.6.3 Stretches Graphs
                                                                                                • 2.6.4 Composite Transformations of Graphs
                                                                                                • 2.7 Polynomial Functions
                                                                                                  • 2.7.1 Factor & Remainder Theorem
                                                                                                    • 2.7.2 Polynomial Division
                                                                                                      • 2.7.3 Polynomial Functions
                                                                                                        • 2.7.4 Roots of Polynomials
                                                                                                        • 2.8 Inequalities
                                                                                                          • 2.8.1 Solving Inequalities Graphically
                                                                                                            • 2.8.2 Polynomial Inequalities
                                                                                                            • 2.9 Further Functions & Graphs
                                                                                                              • 2.9.1 Modulus Functions
                                                                                                                • 2.9.2 Modulus Transformations
                                                                                                                  • 2.9.3 Modulus Equations & Inequalities
                                                                                                                    • 2.9.4 Reciprocal & Square Transformations
                                                                                                                  • 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 Toolkit
                                                                                                                                • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                                                                                  • 3.3.2 Non Right-Angled Trigonometry
                                                                                                                                    • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                                                                                    • 3.4 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 Compound Angle Formulae
                                                                                                                                                    • 3.6.3 Double Angle Formulae
                                                                                                                                                      • 3.6.4 Relationship Between Trigonometric Ratios
                                                                                                                                                        • 3.6.5 Linear Trigonometric Equations
                                                                                                                                                          • 3.6.6 Quadratic Trigonometric Equations
                                                                                                                                                          • 3.7 Inverse & Reciprocal Trig Functions
                                                                                                                                                            • 3.7.1 Reciprocal Trig Functions
                                                                                                                                                              • 3.7.2 Inverse Trig Functions
                                                                                                                                                              • 3.8 Further Trigonometry
                                                                                                                                                                • 3.8.1 Trigonometric Proof
                                                                                                                                                                  • 3.8.2 Strategy for Trigonometric Equations
                                                                                                                                                                  • 3.9 Vector Properties
                                                                                                                                                                    • 3.9.1 Introduction to Vectors
                                                                                                                                                                      • 3.9.2 Position & Displacement Vectors
                                                                                                                                                                        • 3.9.3 Magnitude of a Vector
                                                                                                                                                                          • 3.9.4 The Scalar Product
                                                                                                                                                                            • 3.9.5 Geometric Proof with Vectors
                                                                                                                                                                            • 3.10 Vector Equations of Lines
                                                                                                                                                                              • 3.10.1 Vector Equations of Lines
                                                                                                                                                                                • 3.10.2 Applications to Kinematics
                                                                                                                                                                                  • 3.10.3 Pairs of Lines in 3D
                                                                                                                                                                                    • 3.10.4 The Vector Product
                                                                                                                                                                                      • 3.10.5 Shortest Distances with Lines
                                                                                                                                                                                      • 3.11 Vector Planes
                                                                                                                                                                                        • 3.11.1 Vector Equations of Planes
                                                                                                                                                                                          • 3.11.2 Intersections of Lines & Planes
                                                                                                                                                                                            • 3.11.3 Angles Between Lines & Planes
                                                                                                                                                                                              • 3.11.4 Shortest Distances with Planes
                                                                                                                                                                                            • 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 & Regression
                                                                                                                                                                                                                • 4.3 Probability
                                                                                                                                                                                                                  • 4.3.1 Probability & Types of Events
                                                                                                                                                                                                                    • 4.3.2 Conditional Probability
                                                                                                                                                                                                                      • 4.3.3 Bayes' Theorem
                                                                                                                                                                                                                        • 4.3.4 Sample Space Diagrams
                                                                                                                                                                                                                        • 4.4 Probability Distributions
                                                                                                                                                                                                                          • 4.4.1 Discrete Probability Distributions
                                                                                                                                                                                                                            • 4.4.2 Mean & Variance
                                                                                                                                                                                                                            • 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
                                                                                                                                                                                                                                      • 4.7 Further Probability Distributions
                                                                                                                                                                                                                                        • 4.7.1 Probability Density Function
                                                                                                                                                                                                                                      • 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 Higher 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
                                                                                                                                                                                                                                                                          • 5.7 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.7.1 Basic Limits & Continuity
                                                                                                                                                                                                                                                                            • 5.8 Advanced Differentiation
                                                                                                                                                                                                                                                                              • 5.8.1 First Principles Differentiation
                                                                                                                                                                                                                                                                                • 5.8.2 Applications of Chain Rule
                                                                                                                                                                                                                                                                                  • 5.8.3 Implicit Differentiation
                                                                                                                                                                                                                                                                                    • 5.8.4 Differentiating Further Functions
                                                                                                                                                                                                                                                                                    • 5.9 Advanced Integration
                                                                                                                                                                                                                                                                                      • 5.9.1 Integrating Further Functions
                                                                                                                                                                                                                                                                                        • 5.9.2 Further Techniques of Integration
                                                                                                                                                                                                                                                                                          • 5.9.3 Integrating with Partial Fractions
                                                                                                                                                                                                                                                                                            • 5.9.4 Advanced Applications of Integration
                                                                                                                                                                                                                                                                                              • 5.9.5 Modelling with Volumes of Revolution
                                                                                                                                                                                                                                                                                              • 5.10 Differential Equations
                                                                                                                                                                                                                                                                                                • 5.10.1 Numerical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                  • 5.10.2 Analytical Solutions to Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.10.3 Modelling with Differential Equations
                                                                                                                                                                                                                                                                                                    • 5.11 MacLaurin Series
                                                                                                                                                                                                                                                                                                      • 5.11.1 Maclaurin Series
                                                                                                                                                                                                                                                                                                        • 5.11.2 Maclaurin Series from Differential Equations
                                                                                                                                                                                                                                                                                                        • 5.12 Further Limits (inc l'Hôpital's Rule)
                                                                                                                                                                                                                                                                                                          • 5.12.1 Further Limits
                                                                                                                                                                                                                                                                                                        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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