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This lesson focuses on finding the gradient of a straight line connecting two points, using the example of points (4, -11) and (-2, 7). We will also explore the concept of differentiation to determine the gradient of curves, such as the function y = x². The key idea is to find the gradient of the tangent at a specific point on the curve, which is the essence of differentiation. You will learn how to calculate derivatives using the power rule and apply these concepts to various functions, enhancing your understanding of slopes and rates of change.
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DifferentiationLesson 1 Chapter 7
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)
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 = x2 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
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
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
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
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
