# Chapter one

Mathematical skills

## Section eight: Differentiation

This chapter is on the theme of supporting mathematics that are needed for engineering problem solving, analysis, design and evaluation.

This section gives an introduction to differentiation beginning from a brief discussion of concepts and derivations. Some users will be able to skip this.

There are also a number of worked examples illustrating various common techniques. These include product and quotient rules, implicit differentiation and parametric differentiation.

### 1. Concepts and definitions

This resource introduces the concept of differentiation and the derivative. What do these words mean and what notation is used to represent them? It assumes that viewers are familiar with the terminology of gradient.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 428KB).

### 2. First principle derivations

This derives the derivatives of common functions from first principles. A good understanding of the origins of derivatives will ensure both correct usage and flexibility when something unusual appears.

Skip straight to 'worked examples using a lookup table' to avoid detailed explanations and derivations.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 460KB).

### 3. Superposition

Superposition denotes a scenario where a function is a sum of several simple functions. The word superposition is used to denote that one can treat the functions separately and simply add the result.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 397KB).

### 4. Origins of the product rule

The product rule is to help deal with scenarios where a function is made up as a product of other functions. Therefore, the derivative is not available in a basic table. This resource derives the product rule.

Skip to 'worked examples with the product rule' for worked examples.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 409KB).

### 5. Origins of the quotient rule

The quotient rule is to help deal with scenarios where a function is made up by dividing different functions. Therefore, the derivative is not available in a basic table. This resource is focused on deriving the quotient rule.

Skip to 'worked examples with the quotient rule' for worked examples.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 405KB).

### 6. Worked examples using a lookup table

This section assumes that derivatives of common functions are known and stored in a table. It uses several numerical examples to show how the table can be used to find derivatives of commonly encountered functions.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 491KB).

### 7. Worked examples with the product rule

This section assumes that derivatives of common functions are known and stored in a table. It uses several numerical examples to show how the table in combination with the product rule can be used.

*There is a typing error at 2 minutes 30 seconds, **6x*⁵* is written rather than 6x².*

A talk through video is on YouTube. View the PowerPoint slides (PDF, 408KB).

### 8. Worked examples with the quotient rule

This section assumes derivatives of common functions are known and stored in a table. It uses several numerical examples to show how the table can be used in combination with the quotient rule to find derivatives of commonly encountered functions (eg, sec, cosec and cot).

A talk through video is on YouTube. View the PowerPoint slides (PDF, 416KB).

### 9. Worked examples with the product and quotient rules

This section assumes that derivatives of common functions are known and stored in a table. It uses several numerical examples to show how the table can be used in combination with both the product and the quotient rules to find derivatives.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 417KB).

### 10. Origins of the chain rule

This section introduces the chain rule. The chain rule is how to differentiate a function of a function. This resource gives a concise derivation or background to the rule so that students understand its origins.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 415KB).

### 11. Worked examples with the chain rule

This resource gives a number of worked examples using the chain rule to differentiate moderately complicated functions.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 319KB).

### 12. Implicit differentiation

Implicit differentiation is used where there is not a simple explicit expression for the dependent variable. This resource shows how the chain rule can be used. Examples include logarithms and inverse trigonometric functions.

*There is a typing error around 11 minutes where 2x+3 is written rather than df/dx.*

A talk through video is on YouTube. View the PowerPoint slides (PDF, 457KB).

### 13. Parametric differentiation

This shows how to differentiate curves that are described using parametric equations. It gives a very brief derivation of the origins of the approach based on first principles, followed by several worked examples.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 442KB).