The Chain Rule

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A sound understanding of the Chain Rule is essential to ensure exam success.

To access a wealth of additional free resources by topic please either use the above Search Bar or click on any of the Topic Links found at the bottom of this page as well as on the Home Page HERE.

Passing the fast paced Higher Maths course significantly increases your career opportunities by helping you gain a place on a college/university course, apprenticeship or even landing a job. A ‘good’ pass at Higher Maths will set you up well for the AH Maths Course next year should you be interested. Please do your very best to keep on top of your studies.

For students looking for extra help with the Higher Maths course in 2020/21 you may wish to consider subscribing to the fantastic additional exam focused resources available in the Online Study Pack.

We hope the resources on this website prove useful and wish you the very best of success with your Higher Maths course in 2020/21.

Please find below:

1. About the Chain Rule

2. Differentiation – Worksheets

3. Differentiation – Videos, Theory Guides & Mind Maps

4. Higher Maths Essential Skills 

5. Higher Maths Exam Worksheets by Topic

6. Higher Maths Past & Practice Papers by Topic 

7. Higher Maths Videos, Theory Guides, Mind Maps & Worksheets 

8. Higher Maths Past & Practice Papers 

9. 40 Non-Calculator Higher Maths Questions & Answers

10. 200 Higher Maths Questions & Answers 

11. Practice Exam Papers A to H – Answers Included

12. 264 SQA Exam Multiple Choice Questions & Answers

13. Higher Maths Exam Check Lists 

14. Old Higher Maths Exam Questions by Topic

15. Higher Maths Text Book Solutions

16. Higher Maths Theory Guides 

17. Higher Maths Mind Maps 

18. Higher Maths Practice Unit Assessments – Solutions Included

19. Higher Maths Past Paper Video Solutions

20. Higher Maths Recommended Text Book

21. Exam Focused Online Study Pack – For students looking for a ‘good’ Pass

 

Higher Maths Resources

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1. About the Chain Rule 

The Chain Rule is used in Further Calculus.

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• The n goes to the front of the bracket
• Inside the bracket stays the same ⇒ (ax + b)
• Multiply by the differentiated first bracket ⇒ a
• Simplify by bringing the ‘a’ to the front of the bracket

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Example 1

• f (x) = (3x³ – 2x² + 5)³
• f ‘ (x) = 3 (3x³ – 2x² + 5)² (9x² – 4x)

 

Example 2

• f (x) = sin³x = (sinx)³
• f ‘ (x) = 3(sinx)²(cos x) = 3sin²xcosx

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In the above examples

• The power goes to the front
• The power is then deducted by 1
• The first bracket stays the same
• The second bracket is the first bracket differentiated

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2. Differentiation – Worksheets

Thanks to the SQA and authors for making the excellent resources below freely available. Please use regularly for revision prior to assessments, tests and the final exam.

Clear, easy to follow, step-by-step worked solutions to all worksheets below are available in the Online Study Pack.

Worksheets ____________________	Topic _____________________________	Answers ________
Essential Skills 4	Equation of a Tangent to a Curve	Answers
Essential Skills 5	Stationary Points	Answers
Essential Skills 17	Derived Graphs	Answers
Essential Skills 22	Further Differentiation (Chain Rule)	Answers
Higher Exam Questions	Derived Graphs	Answers
Higher Exam Questions	Differentiation - 1	Answers
Higher Exam Questions	Differentiation - 2	Answers
Higher Exam Questions	Differentiation (Optimisation)	Answers
Higher Exam Questions	Further Calculus	Answers

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3. Differentiation – Videos, Theory Guides & Mind Maps

Thanks to the authors for making the excellent resources below freely available. Please use regularly for revision prior to assessments, tests and the final exam.

Larbert Maths Videos __________________________________	maths180.com Videos ______________________________________	HSN Theory Guides ____________________________	Mind Maps ______________________
Applications of Derivatives	Chain Rule - Increasing/Decreasing	Differentiation Theory (HSN)	Differentiation (HSN)
Chain Rule	Optimisation - More Challenging Examples	Further Calculus Theory (HSN)	Preparing to Differentiate
Closed Intervals	Rate of Change & Tangent to a Curve		Differentiation 1
Complex Differentiation	Trig Functions & Chain Rule		Differentiation 2
Curve Sketching			Further Differentiation
Derived Graphs
Equations of Tangents
Fractions & Roots
Increasing & Decreasing Functions
Introduction to Differentiation
Leibniz Notation
Optimisation
Stationary Points
Sin x & Cos x

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4. Higher Maths Essential Skills 

Thanks to Mr G Rennie for making the excellent resources below freely available. The Essential Skills Worksheets can be used for general revision, homework, consolidation of a topic or preparation for assessments, tests and exams.

Essential Skills _______________	Topic __________________________	Answers ________	Essential Skills _______________	Topic _________________________	Answers ___________
Essential Skills 1	Median of a Triangle	Answers	Essential Skills 18	Logarithmic Equations	Answers
Essential Skills 2	Perpendicular Bisectors	Answers	Essential Skills 19	Proving Trigonometric Identities	Answers
Essential Skills 3	Altitude of a Triangle	Answers	Essential Skills 20	Related Graphs	Answers
Essential Skills 4	Equation of a Tangent to a Curve	Answers	Essential Skills 21	Scalar Product	Answers
Essential Skills 5	Stationary Points	Answers	Essential Skills 22	Further Differentiation	Answers
Essential Skills 6	Quadratic Inequalities	Answers	Essential Skills 23	Further Integration	Answers
Essential Skills 7	Completing the Square	Answers	Essential Skills 24	Recurrence: Consecutive Terms	Answers
Essential Skills 8	Tangent to a Circle	Answers	Essential Skills 25	Differential Equations	Answers
Essential Skills 9	Intersection Lines and Circles	Answers	Essential Skills 26	Definite Integrals	Answers
Essential Skills 10	Section Formula	Answers	Essential Skills 27	Composite Functions	Answers
Essential Skills 11	Trigonometric Formula	Answers	Essential Skills 28	Inverse Functions	Answers
Essential Skills 12	Related Angles	Answers	Essential Skills 29	Gradients and Angles	Answers
Essential Skills 13	Trigonometric Equations	Answers	Essential Skills 30	Angle between Vectors	Answers
Essential Skills 14	Synthetic Division	Answers	Essential Skills 31	Natural Logarithms	Answers
Essential Skills 15	Limit of a Recurrence Relation	Answers	Essential Skills 32	Logs: Connecting 2 Variables	Answers
Essential Skills 16	The Wave Function	Answers	Essential Skills 33	Using the Discriminant	Answers
Essential Skills 17	Derived Graphs	Answers

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5. Higher Maths Exam Worksheets by Topic

Thanks to the SQA and authors for making the excellent resources below freely available. The worksheets by topic are a fantastic study resource since they are actual past paper exam questions.

Clear, easy to follow, step-by-step worked solutions to all new CfE Higher Maths Questions below are available in the Online Study Pack.

Number ______	Topic ___________________________	Answers ____________	Number ______	Topic ___________________________	Answers _____________
1	Circles	Answers	21	Polynomials	Answers
2	Circles (Old Higher)	Ans Included	22	Polynomials (Old Higher)	Ans Included
3	Differentiation - 1	Answers	23	Quadratics	Answers
4	Differentiation - 2	Answers	24	Quadratics (Old Higher)	Ans Included
5	Differentiation (Optimisation)	Answers	25	Recurrence Relations - 1	Answers
6	Differentiation (Old Higher)	Ans Included	26	Recurrence Relations - 2	Answers
7	Exponentials & Logs	Answers	27	Recurrence Relations (Old Higher)	Ans Included
8	Exponentials & Logs (Old Higher)	Ans Included	28	Straight Lines - 1	Answers
9	Functions	Answers	29	Straight Lines - 2	Answers
10	Functions & Graphs	Answers	30	Straight Lines (Old Higher)	Ans Included
11	Functions (Old Higher)	Ans Included	31	Trig Formulae & Equations - 1	Answers
12	Further Calculus	Answers	32	Trig Formulae & Equations - 2	Answers
13	Further Calculus (Old Higher)	Ans Included	33	Trig Addition Formula (Old Higher)	Ans Included
14	Graphs of Functions	Answers	34	Trig Graphs & Eqns (Old Higher)	Ans Included
15	Graphs of Functions (Old Higher)	Ans Included	35	Vectors	Answers
16	Integration - 1	Answers	36	Vectors (Old Higher)	Ans Included
17	Integration - 2	Answers	37	Wave Function	Answers
18	Integration - Area under a Curve	Answers	38	Wave Function (Old Higher)	Ans Included
19	Integration (Old Higher)	Ans Included	39	X-Mas Revision Worksheet	Solutions
20	Polynomials & Quadratics	Answers

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6. Higher Maths Past & Practice Papers by Topic

Thanks to the SQA for making the excellent resources below freely available. Questions and answers have been split up by topic for your ease of reference.

Clear, easy to follow, step-by-step worked solutions to all questions below are available in the Online Study Pack.

. Paper _____________

. Marking __________

. Circles ______

Differentiation & Optimisation ____________

Exp's & Logs ________

Functions & Graphs _________

Further Calculus ________

. Integration _________

Polynomials & Quadratics ____________

Recurrence Relations __________

Straight Lines _________

Trig Formulae & Equations ____________

. Vectors ________

Wave Function _________

2019 P1

Marking P1

Q3,16

Q1

Q14

Q10,12

Q6,11,17b

Q8

Q2

Q4

Q5,7

Q13,15,17a

Q9

2019 P2

Marking P2

Q15

Q7b,11

Q9,12

Q5,8

Q2,13

Q7,10

Q4

Q1

Q6b

Q3,14

Q6a

2018 P1

Marking P1

Q4

Q6,11

Q2,11,15

Q3,14

Q10

Q7

Q1,8

Q13

Q5,9,12

2018 P2

Marking P2

Q5c,12

Q3,9

Q11

Q6

Q1

Q4,7a,10

7b,c

Q5a,b

Q2

Q8

2017 P1

Marking P1

Q2

Q8,15c

Q12

Q1,6,15a

Q3,13

Q10,15b

Q4

Q9

Q7,11

Q5

Q14

2017 P2

Marking P2

Q3,10

Q4,7

Q9

Q2

Q8

Q1

Q6,11

Q5

2016 P1

Marking P1

Q4,8

Q2,9

Q14

Q6,10,12

Q5

Q15

Q3

Q1

Q13

Q7,11

2016 P2

Marking P2

Q4

Q7

Q6

Q10,11b

3b,9

Q2,3a

Q1

Q8b,11a

Q5

Q8a

2015 P1

Marking P1

Q11,14

Q2,7

Q6

Q4,5,13

Q12,15

Q3,8

Q9

Q10

Q1

2015 P2

Marking P2

Q5

Q8

Q2

Q7a

Q4

Q3

Q1

Q7b

Q6

Q9

Specimen P1

Marking P1

Q11

Q8

Q10

Q1

Q4,7

Q2,5,9

Q6,12

Q3

Specimen P2

Marking P2

Q5

Q3b,8

Q4,7

Q9

Q3a

Q2

Q1,6

Q10

Exemplar P1

Marking P1

Q1

Q8,11

Q10

Q3

Q2,5

Q6

Q4,7

Q9

Exemplar P2

Marking P2

Q2

Q9,10

Q7

Q8

Q4

Q3

Q1

Q5,6

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7. Higher Maths Videos, Theory Guides, Mind Maps & Worksheets

Dozens of Higher Maths Videos provide quality lessons by topic. Also included are excellent Theory Guides, Mind Maps and Revision Worksheets with actual Higher Maths exam questions.

Please click on our new Higher Maths Videos & Worksheets by topic dedicated page.

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8. Higher Maths Past & Practice Papers

Thanks to the SQA for making the excellent resources below freely available.

Clear, easy to follow, step-by-step worked solutions to all CfE Higher Papers below are available in the Online Study Pack.

Year _____	Paper Type _________________	Exam Paper ______________	Marking Scheme __________________
2019	Higher (CfE)	Exam Paper	Marking Scheme
2018	Higher (CfE)	Exam Paper	Marking Scheme
2018	Specimen (CfE)	Paper 1, Paper 2	Marking Scheme
2017	Higher (CfE)	Exam Paper	Marking Scheme
2016	Higher (CfE)	Exam Paper	Marking Scheme
2015	Higher (CfE)	Exam Paper	Marking Scheme
2015	Specimen (CfE)	Specimen Paper	Marking Scheme
2015	Exemplar (CfE)	Exemplar Paper	Marking Scheme
2015	Higher (Old)	Exam Paper	Marking Scheme
2014	Higher	Exam Paper	Marking Scheme
2013	Higher	Exam Paper	Marking Scheme
2012	Higher	Exam Paper	Marking Scheme
2011	Higher	Exam Paper	Marking Scheme
2010	Higher	Exam Paper	Marking Scheme
2009	Higher	Exam Paper	Marking Scheme
2008	Higher	Exam Paper	Marking Scheme
2007	Higher	Exam Paper	Marking Scheme
2006	Higher	Exam Paper	Marking Scheme
2005	Higher	Exam Paper	Marking Scheme
2004	Higher	Exam Paper	Marking Scheme
2003	Higher	Exam Paper	Marking Scheme
2002	Higher	Exam Paper	Marking Scheme
2002	Higher (Winter Diet)	Exam Paper
2001	Higher	Exam Paper	Solutions

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9. 40 Non-Calculator Higher Maths Questions & Answers

Thanks to the SQA and authors for making the excellent resources below freely available. Start with these questions to help build your confidence. Once finished, you may like to move onto the 200 Higher Maths Exam Questions in the next section checking your answers as you go. If stuck, always ask your teacher for help as soon as possible.

Clear, easy to follow, step-by-step worked solutions to all 40 questions below are available in the Online Study Pack.

Exam Questions ______________________________________	Answers ___________________________
Whole Booklet for Printing	Answers
Sheet A - 10 Questions	Answers
Sheet B - 10 Questions	Answers
Sheet C - 10 Questions	Answers
Sheet D - 10 Questions	Answers

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10. 200 Higher Maths Questions & Answers

Thanks to the SQA and authors for making the excellent resources below freely available. Please try to do as many questions as possible, checking your answers as you go. If stuck, always ask your teacher for help as soon as possible.

Clear, easy to follow, step-by-step worked solutions to all 200 questions below are available in the Online Study Pack.

Exam Questions _____________________	Answers ___________
Whole Booklet for Printing	Answers
Exam Questions 1 - 20	Answers
Exam Questions 21 - 40	Answers
Exam Questions 41 - 60	Answers
Exam Questions 61 - 80	Answers
Exam Questions 81 - 100	Answers
Exam Questions 101 - 120	Answers
Exam Questions 121 - 140	Answers
Exam Questions 141 - 160	Answers
Exam Questions 161 - 180	Answers
Exam Questions 181 - 200	Answers

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11. Practice Exam Papers A to H – Answers Included

Thanks to the SQA and Larkhall Academy for making the excellent resources below freely available. Please use regularly for revision prior to assessments, tests and the final exam.

Clear, easy to follow, step-by-step worked solutions to Practice Papers A to E are available in the Online Study Pack.

Practice Paper _____________	Paper 1 __________	Paper 2 __________	Papers 1 & 2 ____________	Answers ___________
Paper A	Paper 1	Paper 2	Papers 1 & 2	HERE
Paper B	Paper 1	Paper 2	Papers 1 & 2	HERE
Paper C	Paper 1	Paper 2	Papers 1 & 2	HERE
Paper D	Paper 1	Paper 2	Papers 1 & 2	HERE
Paper E	Paper 1	Paper 2	Papers 1 & 2	HERE
Paper F	Paper 1	Paper 2	Papers 1 & 2	HERE
Paper G	Paper 1	Paper 2	Papers 1 & 2	HERE
Paper H	Paper 1	Paper 2	Papers 1 & 2	HERE

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12. 264 SQA Exam Multiple Choice Questions & Answers 

Thanks to the SQA and authors for making the excellent resources below freely available. Multiple choice are primarily C level questions and a great place to start your revision. If stuck, always ask your teacher for help as soon as possible.

Questions __________	Year ______	Exam Multiple Choice __________________	Exam Multiple Choice __________________	Exam Multiple Choice __________________
1 - 20	2015	Questions Only	Questions & Answers	Answers Only
21 - 40	2014	Questions Only	Questions & Answers	Answers Only
41 - 60	2013	Questions Only	Questions & Answers	Answers Only
61 - 80	2012	Questions Only	Questions & Answers	Answers Only
81 - 100	2011	Questions Only	Questions & Answers	Answers Only
101 - 120	2010	Questions Only	Questions & Answers	Answers Only
121 - 264	Mixed	Questions Only	Questions & Answers	Answers Only

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13. Higher Maths Exam Check Lists

Thanks to the SQA and authors for making the excellent resources below freely available. These are fantastic check lists to assess your Higher Maths knowledge. Please try to use these regularly for revision prior to tests, prelims and the final exam.

Description ___________________________________________________	Link ________	Acknowledgements _________________________
Higher Maths Course Description	HERE
SQA Higher Maths Exam Formulae List	HERE	Courtesy of SQA
Check List 1 - Higher Maths Exam Whole Course	HERE	Courtesy of Zeta Maths
Check List 2 - Higher Maths Formulae List Not Given in Exam	HERE
Check List 3 - Higher Maths Exam Whole Course	HERE
Check List 4 - Higher Maths Exam Trig Revision Sheet	HERE
Check List 5 - One Page Summary Guide Unit 1 (HSN)	HERE	Courtesy of HSN
Check List 6 - One Page Summary Guide Unit 2 (HSN)	HERE	Courtesy of HSN
Check List 7 - One Page Summary Guide Unit 3 (HSN)	HERE	Courtesy of HSN

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14. Old Higher Maths Exam Questions by Topic

Thanks to the SQA for making the excellent resources below freely available. The worksheets by topic are a fantastic additional study resource.

Topic ________	Topic Name ___________________________	Link ________	Notes ___________________
Topic 1	Circles	HERE	Answers Included
Topic 2	Differentiation	HERE	Answers Included
Topic 3	Exponentials & Logarithms	HERE	Answers Included
Topic 4	Functions	HERE	Answers Included
Topic 5	Further Calculus	HERE	Answers Included
Topic 6	Graphs of Functions	HERE	Answers Included
Topic 7	Integration	HERE	Answers Included
Topic 8	Polynomials	HERE	Answers Included
Topic 9	Quadratics	HERE	Answers Included
Topic 10	Recurrence Relations	HERE	Answers Included
Topic 11	Straight Line	HERE	Answers Included
Topic 12	Trig Addition Formulae	HERE	Answers Included
Topic 13	Trig Graphs & Equations	HERE	Answers Included
Topic 14	Vectors	HERE	Answers Included
Topic 15	Wave Function	HERE	Answers Included

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15. Higher Maths Text Book Solutions 

Thanks to the AHS for providing the Heinemann Higher Maths text book solutions below. These will prove extremely useful in helping to progress your Higher Maths knowledge. Please note that there may be the odd arithmetic error.

Topic ______________

Topic Name ____________________

Topic 1

Straight Line

1A

1B

1D

1E

1F

1G

1I

1K

1M

1N

1O

1O

Topic 2

Functions & Graphs 1

2A

2B

2C

2D

2F

2G

2I

Topic 3

Functions & Graphs 2

3A

3C

3K

3N

3O

3P

Topic 4

Recurrence

5A

5B

5B

5C

5D

5F

5H

5I

Topic 5

Differentiation

6A

6C

6D

6E

6F

6G

6H

6I

6J

6L

6M

6N

6O

6P

6Q

6R

6S

Topic 6

Polynomials

7B

7C

7D

7E

7F

7G

7H

7I

7J

Topic 7

Quadratics

8C

8D

8E

8F

8H

8I

8J

8K

Topic 8

Integration

9G

9H

9I

9L

9N

9P

9Q

Topic 9

Trigonometry

11B

11C

11D

11E

11F

11G

11H

Topic 10

Circles

12B

12D

12F

12H

12J

12K

12L

Topic 11

Vectors

13A

13C

13D

13E

13F

13G

13I

13K

13L

13M

13N

13O

13P

13Q

13R

13S

13U

Topic 12

Further Calculus

14B

14C

14C

14E

14G

14H

14I

14J

14K

Topic 13

Exp's & Logs

15C

15D

15E

15F

15G

15H

15I

15I

15J

15K

15L

Topic 14

Wave Function

16A

16C

16D

16E

16F

16G

16H

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16. Higher Maths Theory Guides 

Thanks to HSN for making the excellent Higher Maths Theory Guides freely available for all to use. These will prove a fantastic resource in helping you consolidate your understanding of Higher Maths.

Theory Guides _________________	Topic ____________________________________________	Link _______
Theory Guide 1	All Topics Unit 1 Theory (HSN)	HERE
Theory Guide 2	All Topics Unit 1 - One Page Summary Guide (HSN)	HERE
Theory Guide 3	All Topics Unit 2 Theory (HSN)	HERE
Theory Guide 4	All Topics Unit 2 - One Page Summary Guide (HSN)	HERE
Theory Guide 5	All Topics Unit 3 Theory (HSN)	HERE
Theory Guide 6	All Topics Unit 3 - One Page Summary Guide (HSN)	HERE
Theory Guide 7	All Topics Units 1,2 & 3 Theory (HSN)	HERE
Theory Guide 8	Circles Theory (HSN)	HERE
Theory Guide 9	Differentiation Theory (HSN)	HERE
Theory Guide 10	Exponentials & Logarithms Theory (HSN)	HERE
Theory Guide 11	Functions & Graphs Theory (HSN)	HERE
Theory Guide 12	Further Calculus Theory (HSN)	HERE
Theory Guide 13	Graphs Transformations Theory (Movement & Reflection)	HERE
Theory Guide 14	Graphs Transformations Summary Sheet	HERE
Theory Guide 15	Integration Theory (HSN)	HERE
Theory Guide 16	Polynomials & Quadratics Theory (HSN)	HERE
Theory Guide 17	Sequences Theory (HSN)	HERE
Theory Guide 18	Straight Line Theory (HSN)	HERE
Theory Guide 19	Trigonometry Theory (HSN)	HERE
Theory Guide 19	Vectors Theory (HSN)	HERE
Theory Guide 20	Wave Function Theory (HSN)	HERE

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17. Higher Maths Mind Maps

Thanks to the authors for providing the excellent resources below. These will prove a fantastic resource in helping you prepare for assessments, tests and the final exam.

Mind Map ____________	Topic ____________________________	Mind Map ___________	Topic ____________________________
Mind Map 1	Circle 1	Mind Map 16	Polynomials & Quadratics
Mind Map 2	Circle 2	Mind Map 17	Polynomials
Mind Map 3	Composite Functions	Mind Map 18	Quadratics
Mind Map 4	Compound Angle Formula	Mind Map 19	Recurrence Relations 1
Mind Map 5	Differentiation (Preparation)	Mind Map 20	Recurrence Relations 2
Mind Map 6	Differentiation 1	Mind Map 21	Straight Lines 1
Mind Map 7	Differentiation 2	Mind Map 22	Straight Lines 2
Mind Map 8	Differentiation (Further)	Mind Map 23	Trigonometry 1
Mind Map 9	Functions & Graphs	Mind Map 24	Trigonometry 2
Mind Map 10	Functions	Mind Map 25	Vectors 1
Mind Map 11	Integration (Preparation)	Mind Map 26	Vectors 2
Mind Map 12	Integration 1	Mind Map 27	Vectors 3
Mind Map 13	Integration 2	Mind Map 28	Wave Function 1
Mind Map 14	Logs & Exponentials 1	Mind Map 29	Wave Function 2
Mind Map 15	Logs & Exponentials 2

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18. Higher Maths Practice Unit Assessments – Solutions Included

Thanks to the authors for making the excellent resources below freely available for all to use. Please use regularly for revision prior to assessments, tests and the final exam.

Unit _________	Paper ____________	Solutions ___________
Unit One	Practice A	Solutions
Unit One	Practice B	Solutions
Unit Two	Practice A	Solutions
Unit Two	Practice B	Solutions
Unit Three	Practice A	Solutions
Unit Three	Practice B	Solutions
Unit Four	Practice A	Solutions

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19. Higher Maths Past Paper Video Solutions

Please click DLB Maths to view Higher Maths Past Paper video solutions. This will prove an excellent resource in helping you prepare for assessments, tests and the final exam.

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20. Higher Maths Recommended Text Book

Please find below our highly recommended text book which can be ordered by clicking on the book/link.

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21. Exam Focused Higher Maths Online Study Pack 

Through step-by-step worked solutions to exam questions available in the Online Study Pack we cover everything you need to know about the Chain Rule to pass your final exam.

For students looking for a ‘good’ pass at Higher Maths in 2021 you may wish to consider subscribing to the fantastic additional exam focused resources available in the Online Study Pack. Subscribing to the Online Study Pack may therefore be one of your best ever investments. Please click Online Study Pack  to view screenshots, examples and instructions on how to subscribe.

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