# http://www.mathcentre.ac.uk/resources/uploaded/mc-bus-introtodiff-2009-1.pdf

Introduction to differentiation mc-bus-introtodiff-2009-1 Introduction This leaflet provides a rough and ready introduction to differentiation. This is a technique used to calculate the gradient, or slope, of a graph at different points. The gradient function Given a function, for example, y = x2 , it is possible to derive a formula for the gradient of its graph. We can think of this formula as the gradient function, precisely because it tells us the gradient of the graph. For example, when y = x2 the gradient function is 2x So, the gradient of the graph of y = x2 at any point is twice the x value there. To understand how this formula is actually found you would need to refer to a textbook on calculus. The important point is that using this formula we can calculate the gradient of y = x2 at different points on the graph. For example, when x = 3, the gradient is 2 × 3 = 6. when x = −2, the gradient is 2 × (−2) = −4. How do we interpret these numbers ? A gradient of 6 means that values of y are increasing at the rate of 6 units for every 1 unit increase in x. A gradient of −4 means that values of y are decreasing at a rate of 4 units for every 1 unit increase in x. Note that when x = 0, the gradient is 2 × 0 = 0. Below is a graph of the function y = x2 . Study the graph and you will note that when x = 3 the graph has a positive gradient. When x = −2 the graph has a negative gradient. When x = 0 the gradient of the graph is zero. Note how these properties of the graph can be predicted from knowledge of the gradient function, 2x. y 15 10 When x = -2 the gradient is negative and equal to - 4. - 4 - 3 -2 -1 When x = 3 the gradient is positive and equal to 6 5 0 When x = 0 the gradient is zero. x 1 2 3 4 Example When y = x3 , its gradient function is 3x2 . Calculate the gradient of the graph of y = x3 when a) x = 2, b) x = −1, c) x = 0. www.mathcentre.ac.uk 1 c mathcentre 2009 Solution a) when x = 2 the gradient function is 3(2)2 = 12. b) when x = −1 the gradient function is 3(−1)2 = 3. c) when x = 0 the gradient function is 3(0)2 = 0. Notation for the gradient function You will need to use a notation for the gradient function which is in widespread use. If y is a function of x, that is y = f (x), we write its gradient function as dy . dx dy , pronounced ‘dee y by dee x’, is not a fraction even though it might look like one! This notation dx dy can be confusing. Think of as the ‘symbol’ for the gradient function of y = f (x). The process dx dy is called differentiation with respect to x. of finding dx Example For any value of n, the gradient function of xn is nxn−1 . We write: if y = xn , then dy = nxn−1 dx You have seen specific cases of this result earlier on. For example, if y = x3 , dy = 3x2 . dx More notation and terminology dy , are y ′ , pronounced ‘y dash’, When y = f (x) alternative ways of writing the gradient function, dx df or , or f ′ , pronounced ‘f dash’. In practice you do not need to remember the formulas for the dx gradient functions of all the common functions. Engineers usually refer to a table known as a Table of Derivatives. A derivative is another name for a gradient function. The derivative is also known as the rate of change of a function. Exercises 1. Given that when y = x2 , dy dx = 2x, find the gradient of y = x2 when x = 7. 2. Given that when y = xn , dy dx = nxn−1 , find the gradient of y = x4 when a) x = 2, b) x = −1. 3. Find the rate of change of y = x3 when a) x = −2, 4. Given that when y = 7x2 + 5x, dy dx b) x = 6. = 14x + 5, find the gradient of y = 7x2 + 5x when x = 2. Answers 1. 14. 2. a) 32, b) −4. www.mathcentre.ac.uk 3. a) 12, b) 108. 2 4. 33. c mathcentre 2009

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