1. 11: The Rule for Differentiation © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules

2. The Rule for Differentiation Module C1 AQA Edexcel OCR MEI/OCR Module C2 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

3. The Rule for Differentiation The gradient of a straight line is given by The Gradient of a Straight Line where and are points on the line

4. The Rule for Differentiation 7 - 1 = 6 Solution: 3 - 1 = 2 difference in the x -values difference in the y -values x x e.g. Find the gradient of the line joining the points with coordinates and

5. The Rule for Differentiation The gradient of a straight line is given by We use this idea to get the gradient at a point on a curve This branch of Mathematics is called Calculus Gradients are important as they measure the rate of change of one variable with another. For the graphs in this section, the gradient measures how y changes with x

6. The Rule for Differentiation Tangent at (2, 4) x The Gradient at a point on a Curve Definition: The gradient of a point on a curve equals the gradient of the tangent at that point. e.g. 3 12 The gradient of the tangent at (2, 4) is So, the gradient of the curve at (2, 4) is 4

7. The Rule for Differentiation The gradient changes as we move along a curve e.g.

8. The Rule for Differentiation

14. For the curve we have the following gradients: Point on the curveGradient At every point, the gradient is twice the x -value

15. The Rule for Differentiation Point on the curve For the curve we have the following gradients: Gradient At every point, the gradient is twice the x -value

16. The Rule for Differentiation At every point on the gradient is twice the x -value This rule can be written as The notation comes from the idea of the gradient of a line being d d x y

17. The Rule for Differentiation d d x y At every point on the gradient is twice the x -value This rule can be written as The notation comes from the idea of the gradient of a line being is read as “ dy by dx ” The function giving the gradient of a curve is called the gradient function

18. The Rule for Differentiation The rule for the gradient function of a curve of the form is  “subtract 1 from the power”  “power to the front and multiply” Although this rule won’t be proved, we can illustrate it for by sketching the gradients at points on the curve Other curves and their gradient functions

19. The Rule for Differentiation Gradient of x

20. The Rule for Differentiation Gradient of x x

21. The Rule for Differentiation Gradient of x x x

22. The Rule for Differentiation Gradient of x x x x

23. The Rule for Differentiation Gradient of x x x x x

24. The Rule for Differentiation Gradient of x x x x x x

25. The Rule for Differentiation Gradient of x x x x x x x

26. The Rule for Differentiation x x x x x x x

27. The gradients of the functions and can also be found by the rule but as they represent straight lines we already know their gradients gradient = 0 gradient = 1

28. The Rule for Differentiation Summary of Gradient Functions:

29. The Rule for Differentiation The Gradient Function and Gradient at a Point e.g.1 Find the gradient of the curve at the point (2, 12). At x = 2, the gradient Solution:

30. The Rule for Differentiation Exercises 1. Find the gradient of the curve at the point where x  1 At x = - 1, m = -4 Solution: 2. Find the gradient of the curve at the point Solution: At

31. The Rule for Differentiation  The process of finding the gradient function is called differentiation.  The gradient function is called the derivative. The rule for differentiating can be extended to curves of the form where a is a constant.

32. The Rule for Differentiation tangent at x = 1 More Gradient Functions gradient = 3 e.g. Multiplying by 2 multiplies the gradient by 2

33. The Rule for Differentiation gradient = 6 e.g. Multiplying by 2 multiplies the gradient by 2 tangent at x = 1 gradient = 3

34. The Rule for Differentiation e.g. Multiplying by a constant, multiplies the gradient by that constant

35. The Rule for Differentiation The rule can also be used for sums and differences of terms. e.g.

36. The Rule for Differentiation e.g. Find the gradient at the point where x = 1 on the curve Solution: Differentiating to find the gradient function: When x = 1, gradient m = Using Gradient Functions

37. The Rule for Differentiation SUMMARY  The gradient at a point on a curve is defined as the gradient of the tangent at that point  The process of finding the gradient function is called differentiating  The function that gives the gradient of a curve at any point is called the gradient function  The rule for differentiating terms of the form is “power to the front and multiply” “subtract 1 from the power”

38. The Rule for Differentiation Find the gradients at the given points on the following curves: 1. at the point 2. at the point 3. at the point When Exercises

39. The Rule for Differentiation 5. 4.at ( Multiply out the brackets before using the rule ) (Divide out before using the rule ) at When Find the gradients at the given points on the following curves: Exercises

40. The Rule for Differentiation

41. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

42. The Rule for Differentiation SUMMARY  The gradient at a point on a curve is defined as the gradient of the tangent at that point  The process of finding the gradient function is called differentiating  The function that gives the gradient of a curve at any point is called the gradient function  The rule for differentiating terms of the form is “power to the front and multiply” “subtract 1 from the power”

43. The Rule for Differentiation e.g.

44. The Rule for Differentiation e.g. Find the gradient at the point where x = 1 on the curve Solution: Differentiating to find the gradient function: When x = 1, gradient m = Using Gradient Functions