2. 2-1 The Derivative and the Tangent Line Problem 2-2 Basic Differentiation Rules and Rates of Change 2-3 Product/Quotient Rule and Higher-Order Derivatives 2-4 Chain Rule 2-5 Implicit Differentiation 2-6 Related Rates (SKIP for now) Chapter 2 Differentiation

3. Basic Differentiation Rules Product Rule Quotient Rule Derivative of Trigonometric Functions Higher Order Derivatives 2.3 Product/Quotient Rules and Higher Order Derivatives Day 1 http://www.youtube.com/watch?v=-chXvU4pza4 (Chain, Product, Quotient Rule song :)

4. In function notation: Notice that this is not just the product of two derivatives. 1 st x the derivative of the 2 nd + 2 nd x the derivative of the first. (or vice versa) In words:

5. Example: Find You Try… first function derivative of second function derivative of first function second function FOIL to check answer with Warm Up #3.

6. In function notation: Notice that this is not just the quotient of two derivatives. Bottom x the derivative of the top – top x the derivative of the bottom all over bottom squared. In words:

8. Example: derivative of top function derivative of bottom function top function bottom function squared Redo Warm Up #1 using Quotient Rule. bottom function

9. You Try… http://www.youtube.com/watch?v=-chXvU4pza4 (Chain, Product, Quotient Rule song :)

10. Not every quotient needs to be differentiated by the Quotient Rule.

11. You Try… Differentiate: 1. 4. 2. 5. 3. 6.

12. Solutions #1

13. 2. Same derivative by expanding and using the Power Rule. 1. The Product Rule Solutions #2

14. Product Rule: Find f’(x) for Solutions #4 *The derivative could also be found by performing the multiplication and then finding the derivative.

15. Quotient Rule: Solutions #5 division first and then power rule:

16. Using Derivative Rules Suppose u and v are functions that are differentiable at x = 3, and that u(3) = 5, u’(3) = -7, v(3) = 1, and v’(3)= 4. Find the following at x = 3 :

17. Closure Explain how to use the quotient and product rules to find a derivative.

18. Basic Differentiation Rules Product Rule Quotient Rule Derivative of Trigonometric Functions Higher Order Derivatives 2.3 Product/Quotient Rules and Higher Order Derivatives Day 2

19. Consider the function We could make a graph of the slope: slope Now we connect the dots! The resulting curve is a cosine curve. Derivatives of Trigonometric Functions

20. Proof = 0 = 1

21. Slope of y = cos x The curve y´ = –sin x as the graph of the slopes of the tangents to the curve y = cos x.

22. Proof = 0 = 1

24. Find the derivative of f(x) tan x. Challenge:

25. Find the derivatives of the 3 remaining trig functions.

26. Trig Review: Pythagorean Identities Double Angle Formulas

27. Trig Review Double Angle Formulas

28. Example 2: Find the derivative of Example 1:

29. Ex 3: Differentiate:

30. Given For what values of x does the graph of f have a horizontal tangent? Example 4

31. Example 4 Solution

32. Since sec x is never 0, we see that f’(x) when tan x = 1.  This occurs when x = nπ + π/4, where n is an integer. Example 4 Solution = 0

33. x = nπ + π/4

35. 1.Calculate 1.Find the tangent and normal lines to y = x 2 sin x at x = 3. 3. Find You Try…

36. 1.

37. Find the tangent and normal lines to y = x 2 sin x at x = 3. 2.

38. Higher Derivatives The second derivative of a function f is the derivative of the derivative of f at a point x in the domain of the first derivative. DerivativeNotations Second Third Fourth nth

39. Example 5: Givenfind

40. Example 6 Find if.

41. You Try… 2. Given 3. Compute. 4. find Find the 27th derivative of cos x. Challenge:

42. 1.

43. Givenfind 2.

44. 3.

45. Ex 4: =

46. Find the 27th derivative of cos x.  The first few derivatives of f(x) = cos x are as follows:  Therefore, f (24) (x) = cos x  f (27) (x) = sin x Challenge:

47. Closure Give a way of remembering the derivatives for the six trigonometric functions.