Quiz / Weekly #4 Feedback

1. Quiz / Weekly #4 Feedback -More Effort Needed! -Wording of Problems (derivative, slope at a point, slope of tangent line…) -Product / Quotient Rules!!! -Quiz I:g and II:a -Weekly 7 , 8 , 10

2. The Chain Rule 4.1.1

3. The Chain Rule If y = f (u) is a differentiable function of u and u = g(x) is a differentiable function of x, theny= f (g(x)) is a differentiable function of x and or, equivalently,

4. Identify the inner and outer functions Composite y = f (g(x)) Inner u = g(x) Outery= f (u) 1. 2. 3. 4.

5. The General Power Rule • If , where u is a differentiable function of x and n is a rational number, then or, equivalently,

6. Find the Derivative (two ways…)

7. Find the Derivative

8. Homework • Chain Rule Worksheet

9. Simplifying Chain Rule 4.1.2

10. Factoring Out the Least Powers • Find the Derivative

11. Factoring Out the Least Powers • Find the Derivative

12. Factoring Out the Least Powers • Find the Derivative

13. The Chain Rule (using trig) 4.1.3

14. Find the Derivative

15. Find the Derivative

16. Find the Derivative

17. Trig Tangent Line • Find an equation of the tangent line to the graph of at the point (π, 1). Then determine all values of x in the interval (0, 2π) at which the graph of f has a horizontal tangent.

18. Homework • p.153/ 1-11(O) , 21-39 (O) , 59

19. Implicit Differentiation 4.2

20. Find dy/dx

21. Guidelines for Implicit Differentiation • Differentiate both sides of the equation with respect to x. • Collect all terms involving dy / dx on the left side of the equation and move all other terms to the right side of the equation. • Factor dy / dx out of the left side of the equation. • Solve for dy / dx.

22. Find the derivative

23. Homework • p.162/ 1-19odd, 49, 51

24. Example • Determine the slope of the tangent line to the graph of at the point .

25. Example • Determine the slope of the tangent line to the graph of at the point .

26. Finding the Second Derivative Implicitly

27. Finding the Second Derivative Implicitly

28. Example • Find the tangent and normal line to the graph given by at the point .

29. Homework • p.162/ 21-25odd, 27-30, 31-43odd

30. Inverse Functions 1.5/3.8

31. Definition of Inverse Function • A function g is the inverse function of the function f if for each x in the domain of g. and for each x in the domain of f.

32. Verifying Inverse Functions • Show that the functions are inverse functions of each other. and

33. The Existence of an Inverse Function • A function has an inverse function if and only if it is one-to-one. • If f is strictly monotonic on its entire domain, then it is one-to-one and therefore has an inverse function.

34. Existence of an Inverse Function • Which of the functions has an inverse function?

35. Finding an Inverse • Find the inverse function of .

36. The Derivative of an Inverse Function • Let f be a function that is differentiable on an interval I. If f has an inverse function g, then g is differentiable at any x for which . Moreover,

37. Example • Let . a) What is the value of when x = 3? b) What is the value of when x = 3?

38. Homework • p. 44/ 1-6, 7-23odd, 43 • p. 170/ 28, 29bc

39. Inverse Trigonometric Functions 3.8

40. The Inverse Trigonometric Functions Function

41. Evaluating Inverse Trigonometric Functions • Evaluate each function.

42. Solving an Equation

43. Using Right Triangles a) Given y = arcsin x, where , find cos y. b) Given , find tan y.

44. Homework 3.8 Inverse Trig Review worksheet

45. Derivatives of Inverse Trigonometric Functions • Find

46. Derivatives of Inverse Trigonometric Functions

47. Differentiating Inverse Trigonometric Functions

48. Differentiate and Simplify

49. Homework • p. 170/ 1-27odd, 31ab

50. Derivatives of Exponential and Logarithmic Functions 3.9