# Differentiation resources

### Motivating Mathematics (2)

Faulty Tyres - David Saunders

This mathtutor extension discusses how mathematical modelling using differentiation may be used to determine optimum delivery sizes. This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Millenium Bridge - James Robinson

This mathtutor extension describes the effect of resonance on bridges and how differential equations may be used to calculate the effects. This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

### Practice & Revision (3)

Basic Differentiation - A Refresher

This Refresher Booklet has been designed for students who have studied an AS level course in mathematics. It enables the user to practice basic techniques.

Cwrs Gloywi Calcwlws

A Calculus Refresher.
This booklet revises techniques in calculus (differentiation and integration).
This is a welsh language version

### Quick Reference (5)

Introduction to differentiation 8.1

This leaflet provides a rough and ready introduction to differentiation and gives some common terminology and notation. (Engineering Maths First Aid Kit 8.1) There is an accompanying podcast.

Linearity rules

This leaflet explains how to differentiate the sum of two functions, and constant multiples of functions, using 'linearity rules'. (Engineering Maths First Aid Kit 8.3)

Product and quotient rules

This leaflet states and gives examples of the use of the product and quotient rules for differentiation. (Engineering Maths First Aid Kit 8.4

Table of derivatives

This leaflet provides a table of common functions and their derivatives. (Engineering Maths First Aid Kit 8.2)

The chain rule

The chain rule is used for differentiating a function of a function. This leaflet states and illustrates this rule. (Engineering Maths First Aid Kit 8.5)

### Teach Yourself (14)

Applications of differentiation - maxima and minima

This unit explains how differentiation can be used to locate turning points.
It explains what is meant by a maximum turning point and a minimum turning point.

Differentiating sin(x) and cos(x) from first principles

After reading this text, and/or viewing the video tutorial on this topic,
you should be able to
differentiate the functions sin(x)and cos(x) from first principles.

Differentiation by taking logarithms

In this unit we look at how we can use
logarithms to simplify certain functions before we differentiate them.

Differentiation from first principles

After reading this text, and/or viewing
the video tutorial on this topic, you should be able to
understand the process involved in differentiating from first principles and
differentiate some simple functions from first principles.

Differentiation from first principles (powers of x)

This unit looks at some basic differentiation from first principles, and in particular how to differentiate powers of x.

Differentiation of the logarithm and exponential functions

In this unit the natural logarithm function and the exponential function are differentiated from first principles.

Extending the table of derivatives

This unit extends the basic table and produces a more complete and therefore more useful table.

Parametric differentiation

This unit explains how to differentiate a function defined parametrically.

Tangents and normals

This unit explains how to calculate the equation of the tangent and the normal to a curve at a given point.

The Chain Rule

This teach-yourself workbook explains the chain rule which is used to differentiate a function of a function.

Using a table of derivatives

This unit provides a basic table of some standard derivatives.
Many of the results are derived.

### Test Yourself (12)

Chain Rule examples - Numbas

11 questions on the chain rule.
Numbas resources have been made available under a Creative Commons licence by the School of Mathematics & Statistics at Newcastle University.

Chain rule practice - Numbas

11 Questions on the chain rule. Numbas resources have been made available under a Creative Commons licence by Bill Foster and Christian Perfect, School of Mathematics & Statistics at Newcastle University.

Diagnostic test in basic calculus - Numbas

12 calculus questions, differentiation and integration. Useful for self diagnosis. Numbas resources have been made available under a Creative Commons licence by Bill Foster and Christian Perfect, School of Mathematics & Statistics at Newcastle University.

Diagnostic test in differentiation - Numbas

16 questions: Product Rule, Quotient Rule and Chain Rule. For those that want a thorough testing of their basic differentiation using the standard rules. Numbas resources have been made available under a Creative Commons licence by Bill Foster and Christian Perfect, School of Mathematics & Statistics at Newcastle University.

Diagnostic Test in Differentiation - Numbas

26 questions: Product Rule, Quotient Rule and Chain Rule.
For those that want a thorough testing of their basic differentiation using the standard rules.
Numbas resources have been made available under a Creative Commons licence by the School of Mathematics & Statistics at Newcastle University.

Maths EG

Computer-aided assessment of maths, stats and numeracy from GCSE to undergraduate level 2. These resources have been made available under a Creative Common licence by Martin Greenhow and Abdulrahman Kamavi, Brunel University.

Partial Differentiation Test 01 (DEWIS)

Four questions on partial differentiation. The first two involve finding the first and second partial derivative of a polynomial in x and y. The third involves finding a first partial derivative using the chain rule. The fourth involves finding a second partial derivative involving the product rule. DEWIS resources have been made available under a Creative Commons licence by Rhys Gwynllyw & Karen Henderson, University of the West of England, Bristol.

Product Rule examples - Numbas

10 questions on the product rule.
Numbas resources have been made available under a Creative Commons licence by the School of Mathematics & Statistics at Newcastle University.

Product Rule Practice - Numbas

10 questions on the product rule in differentiation. Numbas resources have been made available under a Creative Commons licence by Bill Foster and Christian Perfect, School of Mathematics & Statistics at Newcastle University.

Quotient Rule examples - Numbas

8 questions on the quotient rule.
Numbas resources have been made available under a Creative Commons licence by the School of Mathematics & Statistics at Newcastle University.

Quotient rule practice - Numbas

8 questions on the quotient rule in differentiation. Numbas resources have been made available under a Creative Commons licence by Bill Foster and Christian Perfect, School of Mathematics & Statistics at Newcastle University.

Using partial derivatives to find stationary points - Numbas

Finding the stationary points of functions of 2 variables. Numbas resources have been made available under a Creative Commons licence by Bill Foster and Christian Perfect, School of Mathematics & Statistics at Newcastle University.

### Third Party Resources (2)

Mathematics Support Materials from the University of Plymouth

The output from this project is a library of portable, interactive, web based support packages to help students learn various mathematical ideas and techniques and to support classroom teaching.

There are support materials on ALGEBRA, GRAPHS, CALCULUS, and much more.

This material is offered through the mathcentre site courtesy of Dr Martin Lavelle and Dr Robin Horan from the University of Plymouth.

**Support material from the University of Plymouth:**

The output from this project is a library of portable, interactive, web based support packages to help students learn various mathematical ideas and techniques and to support classroom teaching.

There are support materials on ALGEBRA, GRAPHS, CALCULUS, and much more.

This material is offered through the mathcentre site courtesy of Dr Martin Lavelle and Dr Robin Horan from the University of Plymouth.

University of East Anglia (UEA) Interactive Mathematics and Statistics Resources

The Learning Enhancement Team at the University of East Anglia (UEA) has developed la series of interactive resources accessible via Prezi mind maps : Steps into Numeracy, Steps into Algebra, Steps into Trigonometry, Bridging between Algebra and Calculus, Steps into Calculus, Steps into Differential Equations, Steps into Statistics and Other Essential Skills.

### Video (15)

Differentiating sin(x) and cos(x) from first principles

In this unit we show how to differentiate the sine and cosine functions
from first principles. (Mathtutor Video Tutorial).
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Differentiation by taking logs

In this unit we look at how we can use logarithms to simplify certain functions before we differentiate them. (Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Differentiation from first principles

In this unit we start to explain how differentiation works. The process is known as differentiation from first principles.
(Mathtutor Video Tutorials)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Differentiation from first principles (powers of x)

In this unit we explain how to differentiate powers of x from first principles. (Mathtutor Video Tutorials)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Differentiation of the logarithm and exponential functions

This unit gives details of how logarithmic functions and exponential functions are differentiated from first principles. (Mathtutor Video Tutorials)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Extending the table of derivatives

In this unit we continue to build up The Table of Derivatives using rules described in other units. (Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Implicit Differentiation

Sometimes functions are given not in the form y=f(x) but in a more complicated form in which it is difficult or impossible to express y explicitly in terms of x.. Such functions are called implicit functions. In this unit we explain how these can be differentiated using implicit differentiation. (Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Introduction to differentiation 8.1 - Podcast

Podcast to accompany the Quick Reference Engineering Maths First Aid Kit leaflet 'Introduction to differentiation 8.1' submitted under Creative Commons Licence BY-NC-SA to the mathcentre Community Project by Ciaran Mac an Bhaird, National University of Ireland Maynooth and reviewed by Ann O'Shea, National University of Ireland Maynooth.

Maxima and Minima

In this unit we show how differentiation can be used to find the maximum and minimum values of a function. Because the derivative provides information about
the gradient or slope of the graph of a function we can use it to locate points on a
graph where the gradient is zero. We shall see that such points are often associated
with the largest or smallest values of the function, at least in their immediate
locality. In many applications, a scientist, engineer, or economist for example, will
be interested in such points for obvious reasons such as maximising power, or profit, or minimising losses or costs. (Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Parametric differentiation

Instead of a function y(x) being defined explicitly in terms of the independent variable x, it is sometimes useful to
define both x and y in terms of a third variable, t say, known as a parameter. In this unit we explain how such
functions can be differentiated using a process known as parametric differentiation. (Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Tangents and Normals

This unit explains how differentiation can be used to calculate the equations of the tangent and normal to a curve.
The tangent is a straight line which just touches the curve at a given point. The normal is a straight line which is
perpendicular to the tangent. (Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

The Chain Rule

A special rule, the chain rule, exists for differentiating a function of another function. This unit illustrates this rule. (Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

The Product Rule

A special rule, the product rule, exists for differentiating products of two (or more) functions. This unit illustrates this rule. (Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

The Quotient Rule

A special rule, the quotient rule, exists for differentiating quotients of
two functions. This unit illustrates
this rule. (Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Using a table of derivatives

In this unit we construct a Table of Derivatives of commonly occurring functions. This is done using the knowledge gained in previous units on differentiation from first principles. (Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

### Video with captions (2)

Differentiation from first principles

In this unit we start to explain how differentiation works. The process is known as differentiation from first principles. (Mathtutor Video Tutorials) The video is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Differentiation from first principles (powers of x)

In this unit we explain how to differentiate powers of x from first principles. (Mathtutor Video Tutorials)
The video is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

### Video with captions which require edits (13)

Differentiating sin(x) and cos(x) from first principles

In this unit we show how to differentiate the sine and cosine functions
from first principles. (Mathtutor Video Tutorial).
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Differentiation by taking logs

In this unit we look at how we can use logarithms to simplify certain functions before we differentiate them. (Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Differentiation of the logarithm and exponential functions

This unit gives details of how logarithmic functions and exponential functions are differentiated from first principles. (Mathtutor Video Tutorials)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Extending the table of derivatives

In this unit we continue to build up The Table of Derivatives using rules described in other units. (Mathtutor Video Tutorial)
The video is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Implicit Differentiation

Sometimes functions are given not in the form y=f(x) but in a more complicated form in which it is difficult or impossible to express y explicitly in terms of x.. Such functions are called implicit functions. In this unit we explain how these can be differentiated using implicit differentiation. (Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Introduction to differentiation 8.1 - Podcast

Podcast to accompany the Quick Reference Engineering Maths First Aid Kit leaflet 'Introduction to differentiation 8.1' submitted under Creative Commons Licence BY-NC-SA to the mathcentre Community Project by Ciaran Mac an Bhaird, National University of Ireland Maynooth and reviewed by Ann O'Shea, National University of Ireland Maynooth.

Maxima and Minima

In this unit we show how differentiation can be used to find the maximum and minimum values of a function. Because the derivative provides information about
the gradient or slope of the graph of a function we can use it to locate points on a
graph where the gradient is zero. We shall see that such points are often associated
with the largest or smallest values of the function, at least in their immediate
locality. In many applications, a scientist, engineer, or economist for example, will
be interested in such points for obvious reasons such as maximising power, or profit, or minimising losses or costs. (Mathtutor Video Tutorial)
The video is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Parametric differentiation

Instead of a function y(x) being defined explicitly in terms of the independent variable x, it is sometimes useful to
define both x and y in terms of a third variable, t say, known as a parameter. In this unit we explain how such
functions can be differentiated using a process known as parametric differentiation. (Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Tangents and Normals

This unit explains how differentiation can be used to calculate the equations of the tangent and normal to a curve.
The tangent is a straight line which just touches the curve at a given point. The normal is a straight line which is
perpendicular to the tangent. (Mathtutor Video Tutorial)
The video is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

The Chain Rule

A special rule, the chain rule, exists for differentiating a function of another function. This unit illustrates this rule. (Mathtutor Video Tutorial)
The video is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

The Product Rule

A special rule, the product rule, exists for differentiating products of two (or more) functions. This unit illustrates this rule. (Mathtutor Video Tutorial)
The video is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

The Quotient Rule

A special rule, the quotient rule, exists for differentiating quotients of
two functions. This unit illustrates
this rule. (Mathtutor Video Tutorial)
The video is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Using a table of derivatives

In this unit we construct a Table of Derivatives of commonly occurring functions. This is done using the knowledge gained in previous units on differentiation from first principles. (Mathtutor Video Tutorial)
The video is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.