Differentiation Lesson 1

1. DifferentiationLesson 1 Chapter 7

2. Gradient = We need to be able to find the gradient of a straight line joining two points: Find the gradient of the line joining (4, -11) and (-2, 7)

3. Describe the way the gradient is changing on these graphs:

4. Finding the gradient of a curve (differentiation)

5. How can you find the gradient of a curve if it keeps changing??

6. E.g. the function y = x2 Go to GSP file

7. The ideal way to find a gradient of a curve is to find the gradient of the tangent at the point we are interested in

8. Finding the gradient • The process of finding the gradient of a curve is called “differentiation” • You can differentiate any function to find its gradient

9. The function The derivative (or differential) of the function In general it can be shown that If f(x) = xn where n is a real number Then f ’(x) = nxn-1 f(x) = xn

10. The function The derivative (or differential) of the function. This is the gradient function In general it can be shown that f(x) = xn f ’(x) = nxn-1

11. f(x) = xn f ’(x) = nxn-1 In other words… E.g. 1 -1 1) Multiply function by power of x 2) Subtract 1 from the power

12. f(x) = xn f ’(x) = nxn-1 In other words… E.g. 1 1) Multiply function by power of x 2) Subtract 1 from the power

13. Notation

14. Now do Ex 7B on page 109