1. The Product Rule In words: “Keep the first, differentiate the second” + “Keep the second, differentiate the first”

2. Examples:1.Differentiate

3. Examples:2.Differentiate Now watch this.

4. Examples:3.Differentiate Try this using “words”

6. The Quotient Rule In words: “Keep the denominator, differentiate the numerator” “Keep the numerator, differentiate the denominator” – Denominator 2

7. Examples:1.Differentiate

8. Examples:2.Differentiate Try this using “words”

9. Add a denominator here

10. Derivatives of New Functions Definitions: Reminder: continue

11.     

12.     

13.     

14. Derivative of Use the Quotient Rule now Proof:

15. Derivatives of Prove these and keep with your notes. Use chain rule or quotient rule

16. Example: Given thatshow that

17. Exponential and Logarithmic Functions Reminder: andare inverse to each other. They are perhaps the most important functions in the applications of calculus in the real world. Alternative notation: Two very useful results: Also: Practise changing from exp to log and vice-versa. Learn these!

18.     x y

19. Derivatives of the Exponential and Logarithmic Functions Proof of (ii) (i)(ii)

20. Examples:1.Differentiate Use the Chain Rule 2.Differentiate Use the Product Rule

21. 3.Differentiate Use the Chain Rule 4.Differentiate Use the Quotient Rule

22. Note:In general Useful for reverse i.e. INTEGRATION

23. Higher Derivatives Given that f is differentiable, if is also differentiable then its derivative is denoted by. The two notations are: function 1 st derivative 2 nd derivative …… n th derivative f ……

24. Example: If, write down is first second and third derivatives and hence make a conjecture about its nth derivative. Conjecture:The n th derivative is

25. Rectilinear Motion If displacement from the origin is a function of time I.e. then v - velocity a - acceleration

26. Example:A body is moving in a straight line, so that after t seconds its displacement x metres from a fixed point O, is given by (a)Find the initial dislacement, velocity and acceleration of the body. (b)Find the time at which the body is instantaneously at rest.

27. Extreme Values of a Function Understand the following terms: Critical Points Local Extreme Values  Local maximum  Local minimum End Point Extreme Values  End Point maximum  End Point minimum See, MIA Mathematics 1, Pages 54 – 55

28. A B Consider maximum turning point A. Notice, gradient of for x in the neighbourhood of A is negative. The Nature of Stationary Points i.e. is negative Similarly, gradient of for x in the neighbourhood of B is positive. i.e. is positive Consider a curve and the corresponding gradient function

29. The Nature of Stationary Points Rule for Stationary Points and minimum turning point and maximum turning point and possibly a point of inflexion but must check using a table of signs

30. Example: Consider At S.P. Now what does look like? Notice no Point of Inflexion.

31. Global Extreme Values Understand the following terms: Global Extreme Values  Global maximum  Global minimum See, MIA Mathematics 1, Pages 58 – 59

32. Example: Find the coordinates and nature of the stationary point on the curve At S.P. is a Minimum Turning Point What does this curve look like?

33. x y

34. Optimisation Problems A sector of a circle with radius r cm has an area of 16 cm 2. Show that the perimeter P cm of the sector is given by (a)(a) (b)(b)Find the minimum value of P. (a)(a) l r r now

35. (b)(b) At SP r = 4 gives a minimum stationary value of