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Higher Unit 3

Higher Unit 3. Differentiation The Chain Rule. Further Differentiation Trig Functions. Further Integration . Integrating Trig Functions. The Chain Rule for Differentiating. To differentiate composite functions (such as functions with brackets in them) we can use:.

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Higher Unit 3

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

  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. 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. 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 Back Next Quit

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

  19. Calculus Revision Differentiate Chain Rule Simplify Back Next Quit

  20. Calculus Revision Differentiate Chain Rule Simplify Back Next Quit

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

  22. Calculus Revision 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. 1. Differentiate outside the bracket. 2. Keep the bracket the same. 3. Differentiate inside the bracket. The Chain Rule for DifferentiatingTrig Functions Worked Example:

  27. The Chain Rule for DifferentiatingTrig Functions Example

  28. The Chain Rule for DifferentiatingTrig Functions Example

  29. Calculus Revision Differentiate Back Next Quit

  30. Calculus Revision Differentiate Back Next Quit

  31. Calculus Revision Differentiate Back Next Quit

  32. Calculus Revision Differentiate Back Next Quit

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

  34. Calculus Revision Differentiate Chain Rule Simplify Back Next Quit

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

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

  41. 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. 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 Integrate Standard Integral (from Chain Rule) Back Next Quit

  46. Calculus Revision Integrate Straight line form Back Next Quit

  47. Calculus Revision Use standard Integral (from chain rule) Find Back Next Quit

  48. Calculus Revision Integrate Straight line form Back Next Quit

  49. Calculus Revision Use standard Integral (from chain rule) Find Back Next Quit

  50. Calculus Revision Use standard Integral (from chain rule) Evaluate Back Next Quit

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