Introduction to differentiation

Introduction

Please note: a Statement of Participation is not issued on this course.

This free OpenLearn course, Introduction to differentiation, is an extract from the Open University module MST124 Essential mathematics 1 [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] , an entry level module intended to provide a foundation in the essential mathematical ideas and techniques that underpin the study of mathematics and mathematical subjects such as physics, engineering and economics. MST124 looks at a variety of mathematical topics such as algebra, graphs, trigonometry, coordinate geometry, vectors, differentiation, integration, matrices, complex numbers and associated techniques. It also helps develop the abilities to study mathematics independently, to solve mathematical problems and to communicate mathematics. MST124 follows on naturally from the Open University module, MU123 Discovering mathematics.

Introduction to differentiation consists of material from MST124 Unit 6, Differentiation and has five sections in total. You should set aside between three to four hours to study each of the sections; the whole extract should take about 18 hours to study. The extract is a small part (around 8%) of a large module that is studied over eight months, and so can give only an approximate indication of the level and content of the full module.

The extract, which contains the first introduction to differentiation in MST124, is relatively self-contained and should be reasonably easy to understand for someone who has not studied any of the previous texts in the module. A few techniques and definitions taught in earlier units in MST124 are present in the extract without explanation, therefore a fluency with basic algebra and graphs is essential for this extract.

Mathematical/statistical content at the Open University is usually provided to students in printed books, with PDFs of the same online. This format ensures that mathematical notation is presented accurately and clearly. The PDF of this extract thus shows the content exactly as it would be seen by an Open University student. However, the extract isn't entirely representative of the module materials, because there are no explicit references to use of the MU124 software or to video material (although please note that the PDF may contain references to other parts of MU124). In this extract, some illustrations have also been removed due to copyright restrictions.

Regrettably mathematical and statistical content in PDF files is unlikely to be accessible using a screenreader, and some OpenLearn units may have PDF files that are not searchable. You may need additional help to read these documents.

Section 1 introduces you to the basic ideas of differentiation, by looking at gradients of graphs. Building on these ideas leads to differentiation from first principles. Two different notations are introduced, Lagrange and Leibnitz.

Section 2 looks at finding derivatives of simple functions and introduces you to the constant multiple and sum rules.

Section 3 introduces you to the idea of rates of change by looking at real life situations involving rates of change, namely ‘displacement and velocity’ and ‘total and marginal costs’.

Section 4 looks at using the derivative of a function to deduce useful facts about its graph. You will be introduced to the ideas of stationary points and using the first derivative test to provide information needed to draw a sketch of a function.

Section 5 covers differentiating twice and the second derivative test, used to determine the nature of stationary points. The section ends by looking at rates of change, including the real-life situation of using differentiation to find acceleration.