Differentiation Lesson 1 Chapter 7

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

Describe the way the gradient is changing on these graphs:

Finding the gradient of a curve (differentiation)

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

E.g. the function y = x 2 Go to GSP file

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

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

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

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

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

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

Notation

Now do Ex 7B on page 109