1. Higher Unit 3 Differentiation The Chain Rule
Further Differentiation Trig Functions
Further Integration
Integrating Trig Functions

2. The Chain Rule for Differentiating
To differentiate composite functions (such as functions with brackets in them) we can use:
Example

3. The Chain Rule for Differentiating
You have 1 minute to come up with the rule.
1. Differentiate outside the bracket.
2. Keep the bracket the same.
3. Differentiate inside the bracket.
Good News !
There is an easier way.

4. The Chain Rule for Differentiating
1. Differentiate outside the bracket.
2. Keep the bracket the same.
3. Differentiate inside the bracket.
The Chain Rule for Differentiating
Example
You are expected to do the chain rule all at once

5. The Chain Rule for Differentiating
1. Differentiate outside the bracket.
2. Keep the bracket the same.
3. Differentiate inside the bracket.
The Chain Rule for Differentiating
Example

6. The Chain Rule for Differentiating
Example

7. The Chain Rule for Differentiating Functions
Example
The slope of the tangent is given by the derivative of the equation.
Re-arrange:
Use the chain rule:
Where x = 3:

8. The Chain Rule for Differentiating Functions
Remember
y - b = m(x – a)
Is the required equation

9. The Chain Rule for Differentiating Functions
Example
In a small factory the cost, C, in pounds of assembling x components in a month is given by:
Calculate the minimum cost of production in any month, and the corresponding number of components that are required to be assembled.
Re-arrange

10. The Chain Rule for Differentiating Functions
Using chain rule

11. The Chain Rule for Differentiating Functions
Is x = 5 a minimum in the (complicated) graph?
Is this a minimum?
For x < 5 we have (+ve)(+ve)(-ve) = (-ve)
For x = 5 we have (+ve)(+ve)(0) = 0
x = 5
For x > 5 we have (+ve)(+ve)(+ve) = (+ve)
Therefore x = 5 is a minimum

12. The Chain Rule for Differentiating Functions
The cost of production:
Expensive components? Aeroplane parts maybe ?

13. Calculus Revision
Differentiate
Chain rule
Simplify
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Quit

14. Calculus Revision
Differentiate
Chain Rule
Simplify
Back
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Quit

15. Calculus Revision
Differentiate
Chain Rule
Back
Next
Quit

16. Calculus Revision
Differentiate
Chain Rule
Simplify
Back
Next
Quit

17. Calculus Revision
Differentiate
Chain Rule
Simplify
Back
Next
Quit

18. Calculus Revision Back Next Quit Differentiate Straight line form
Chain Rule
Simplify
Back
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Quit

19. Calculus Revision
Differentiate
Chain Rule
Simplify
Back
Next
Quit

20. Calculus Revision
Differentiate
Chain Rule
Simplify
Back
Next
Quit

21. Calculus Revision Back Next Quit Differentiate Straight line form
Chain Rule
Simplify
Back
Next
Quit

22. Calculus Revision Back Next Quit Differentiate Straight line form
Chain Rule
Simplify
Back
Next
Quit

23. Trig Function Differentiation
The Derivatives of sin x & cos x

24. Trig Function Differentiation
Example

25. Trig Function Differentiation
Example
Simplify expression - where possible
Restore the original form of expression

26. The Chain Rule for Differentiating Trig Functions
1. Differentiate outside the bracket.
2. Keep the bracket the same.
3. Differentiate inside the bracket.
The Chain Rule for Differentiating Trig Functions
Worked Example:

27. The Chain Rule for Differentiating Trig Functions
Example

28. The Chain Rule for Differentiating Trig Functions
Example

29. Calculus Revision
Differentiate
Back
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30. Calculus Revision
Differentiate
Back
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Quit

31. Calculus Revision
Differentiate
Back
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32. Calculus Revision
Differentiate
Back
Next
Quit

33. Calculus Revision Back Next Quit Differentiate Straight line form
Chain Rule
Simplify
Back
Next
Quit

34. Calculus Revision
Differentiate
Chain Rule
Simplify
Back
Next
Quit

35. Calculus Revision Back Next Quit Differentiate Straight line form
Chain Rule
Simplify
Back
Next
Quit

36. Calculus Revision
Differentiate
Back
Next
Quit

37. Calculus Revision
Differentiate
Chain Rule
Simplify
Back
Next
Quit

38. Calculus Revision
Differentiate
Chain Rule
Simplify
Back
Next
Quit

39. Harder integration Integrating Composite Functions we get
You have 1 minute to come up with the rule.
Integrating Composite Functions
Harder integration
we get

40. Integrating Composite Functions
1. Add one to the power.
2. Divide by new power.
3. Compensate for bracket.
Integrating Composite Functions
Example :

41. Integrating Composite Functions
1. Add one to the power.
2. Divide by new power.
3. Compensate for bracket.
Integrating Composite Functions
Example
You are expected to do the integration rule all at once

42. Integrating Composite Functions
Example

43. Integrating Composite Functions
Example

44. Integrating Functions
1. Add one to the power.
2. Divide by new power.
3. Compensate for bracket.
Integrating Functions
Example
Integrating
So we have:
Giving:

45. Calculus Revision Standard Integral (from Chain Rule) Back Next Quit
Integrate
Standard Integral
(from Chain Rule)
Back
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46. Calculus Revision
Integrate
Straight line form
Back
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Quit

47. Calculus Revision Use standard Integral (from chain rule) Back Next
Find
Back
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48. Calculus Revision
Integrate
Straight line form
Back
Next
Quit

49. Calculus Revision Use standard Integral (from chain rule) Back Next
Find
Back
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50. Calculus Revision Use standard Integral (from chain rule) Back Next
Evaluate
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51. Calculus Revision
Evaluate
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52. Calculus Revision
Find p, given
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53. passes through the point (–1, 2).
Calculus Revision
A curve for which
passes through the point (–1, 2).
Express y in terms of x.
Use the point
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54. Given the acceleration a is:
Calculus Revision
Given the acceleration a is:
If it starts at rest, find an expression for the velocity v where
Starts at rest, so
v = 0, when t = 0
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Quit

55. Integrating Trig Functions
Integration is opposite of differentiation
Worked Example

56. Integrating Trig Functions
Integrate outside the bracket
Keep the bracket the same
Compensate for inside the bracket.
Integrating Trig Functions
Special Trigonometry Integrals are
Worked Example

57. Integrating Trig Functions
Integrate outside the bracket
Keep the bracket the same
Compensate for inside the bracket.
Integrating Trig Functions
Example
Break up into two easier integrals
Integrate

58. Integrating Trig Functions
Integrate outside the bracket
Keep the bracket the same
Compensate for inside the bracket.
Integrating Trig Functions
Example
Integrate
Re-arrange

59. Integrating Trig Functions (Area)
Example
The diagram shows the graphs of y = -sin x and y = cos x
Find the coordinates of A
Hence find the shaded area C A S T 0o
180o
270o
90o

60. Integrating Trig Functions (Area)

61. Integrating Trig Functions
Example
Remember cos(x + y) =

62. Integrating Trig Functions

63. Calculus Revision
Find
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64. Calculus Revision
Find
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65. Calculus Revision
Find
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66. Calculus Revision
Integrate
Integrate term by term
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67. Calculus Revision
Find
Integrate term by term
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68. Calculus Revision
Find
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69. passes through the point
Calculus Revision
passes through the point
The curve
Find f(x)
use the given point
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70. passes through the point
Calculus Revision
passes through the point If
express y in terms of x.
Use the point
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Quit

71. passes through the point
Calculus Revision
A curve for which
passes through the point
Find y in terms of x.
Use the point
Back
Next
Quit

72. Are you on Target ! Update you log book
Make sure you complete and correct
ALL of the Calculus questions in the
past paper booklet.