1. AS Maths Masterclass Lesson 3: Introduction to differentiation

2. Learning objectives The student should be able to: understand the meaning of the derivative of a function; apply the derivative rule to simple polynomials; appreciate the first principles approach to differentiation;

3. What is differentiation about and why do we need it? It is all about how the tangent changes as we move along a curve. We can find the maximum and minimum values of a function that constantly changes: This is important when we need to know the rate at which things are changing.

4. Obtaining the derived function You are about to view a webpage. Notice how the tangent line changes as we move along the curve. Click here to see the trace of a tangent In the following webpage we can actually see the value of the slope of the curve marked on a graph as we move along the curve. Click here to see a trace of the derived function

5. Observing the derived function Click here to see a graph of f(x) and its derivative: f ' (x) Now it’s time to search for a pattern. Let’s start by exploring the derived function for Click here for an experiment to find the gradient of f(x) = x^3

6. Establishing a formula We are now in a position to look for a formula for the derivative of f (x). The spreadsheet below is set up for Feel free to experiment by drawing tangents at different points x = a, and changing the steplength h. (See the spreadsheet notes for full details). Try and find the formula for the derived function when Click here for the differentiation spreadsheet.

7. Using the formula Given a function then the derived function is given by Click for basic questions on using the formula Click for questions using the rule on polynomials Click for questions using the rule with one bracket Click for questions with negative powers Click for questions with rational negative powers

8. A first principles approach We now look at how to prove our assertions of the derived function for individual cases. In general, the derivative f ’ (x) evaluated at x = a can be defined as Click here to see how this works with quadratics

9. Practice makes perfect ! Click here to practice first principles on quadratics Click here for first principles applied to rational expressions Click here to apply first principles to surdal expressions Click here for a check test on first principles