1. h Let f be a function such that lim f(2 + h) - f(2) = 5.
Warm-Up
Let f be a function such that
lim f(2 + h) - f(2) = 5.
Which of the following must be true?
h 0 h
I. f is continuous at x = 2
II. f is differentiable at x = 2
III. The derivative of f is continuous at x = 2
(a) I only (b) II only
(c) I and II only (d) I and III only
(e) II and III only

7. Problem of the Day Let f be a differentiable function such that
f(3) = 2 and f '(3) = 5. If the tangent line to the graph of f at x = 3 is used to find an approximation to a zero of f, that approximation is
A) 0.4
B) 0.5
C) 2.6
D) 3.4
E) 5.5

8. Problem of the Day Let f be a differentiable function such that
f(3) = 2 and f '(3) = 5. If the tangent line to the graph of f at x = 3 is used to find an approximation to a zero of f, that approximation is
A) 0.4
B) 0.5
C) 2.6
D) 3.4
E) 5.5
Point (3, 2)
Slope = 5
Tangent y - 2 = 5(x - 3)
Thus y = 5x - 13
To find zero 0 = 5x - 13
x = 2.6

9. 2-3: Product / Quotient Rules & Other Derivatives
Objectives:
Learn and use the product & quotient rules.
Derive derivatives of trignometric functions.
Use higher-order derivatives.
©2002 Roy L. Gover (

10. Important Idea
The derivative of the product is not the product of the derivatives.
𝑇𝑟𝑦 𝑓 𝑥 =2 𝑎𝑛𝑑 𝑔 𝑥 =𝑥

11. If h(x) = f(x)g(x) what is the derivative?

12. If h(x) = f(x)g(x) what is the derivative?
lim
Δx 0
f(x + Δx)g(x + Δx) - f(x)g(x) Δx
add a well chosen zero
f(x+Δx)g(x+Δx) + f(x+Δx)g(x) - f(x+Δx)g(x) - f(x)g(x) Δx

13. If h(x) = f(x)g(x) what is the derivative?
lim
Δx 0
f(x + Δx)g(x + Δx) - f(x)g(x) Δx
add a well chosen zero
f(x+Δx)g(x+Δx) - f(x+Δx)g(x) + f(x+Δx)g(x) - f(x)g(x) Δx
lim
Δx 0
f(x+Δx)(g(x+Δx) - g(x)) + g(x)(f(x+Δx) - f(x)) Δx

14. If h(x) = f(x)g(x) what is the derivative?
lim
Δx 0
f(x + Δx)g(x + Δx) - f(x)g(x) Δx
add a well chosen zero
f(x+Δx)g(x+Δx) - f(x+Δx)g(x) + f(x+Δx)g(x) - f(x)g(x) Δx
lim
Δx 0
f(x+Δx)(g(x+Δx) - g(x)) + g(x)(f(x+Δx) - f(x)) Δx
lim
Δx 0
g(x)(f(x+Δx) - f(x)) Δx
lim
Δx 0
f(x+Δx)(g(x+Δx) - g(x)) + Δx

15. If h(x) = f(x)g(x) what is the derivative?
lim
Δx 0
f(x+Δx) (g(x+Δx) - g(x)) +
lim
Δx 0
g(x) (f(x+Δx) - f(x)) Δx Δx
Evaluate limits
f(x) g'(x) + g(x) f '(x)

16. Product Rule If h(x) = f(x)g(x) what is the derivative?
f(x) g'(x) + g(x) f '(x)
1st times derivative of second +
2nd times derivative of 1st
(Rule extends to cover more than 2 factors)
if j(x) = f(x)g(x)h(x) then
j'(x) = f '(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x)

17. The Product Rule
Memorize

18. The Product Rule
The derivative of the product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first.

19. Example Find the derivative, if it exists:
The product of two functions

20. Example
Find the derivative, if it exists:

21. Important Idea Be sure you simplify your answers by at least:
combining like terms
eliminating negative exponents

22. Try This Find the derivative:
Can you use a method other than the product rule?

23. Example Find the derivative:
In this example, you must use the product rule.
Note the new notation for derivative

24. Try This
Find the derivative:

25. Important Idea
The derivative of a quotient is not the quotient of the derivatives.

26. The Quotient Rule
The derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator all divided by the denominator squared.

27. The Quotient Rule
lo D hi minus hi D lo over lo2

28. We’ll prove this one next week…

29. Example Find the derivative using the quotient rule:
Is there an easier way…

30. Example Find the derivative using the quotient rule:
Is there an easier way…

31. Important Idea
Sometimes it is easier to re-write the function and find the derivative using rules other than the quotient rule.

32. Warm-Up
Find the derivative using the quotient rule and simplify your answer:

33. Warm-Up
Find the derivative using the quotient rule and simplify your answer:

34. Example
Must use the quotient rule on this one…

35. Warm-Up
Find the derivative (hint: re-write and use the quotient rule):

36. Warm-Up
Find the derivative (hint: re-write and use the quotient rule):

37. Try This
Find the derivative (hint: re-write and use the quotient rule):

38. Try This
Find an equation of the line tangent to s(t) at t=2:

39. Example
Find:

40. Try This
Find:
Hint: write and use the quotient rule.

41. Do This
Memorize the derivatives of
It’s on page 123…

42. Try This
Differentiate:
What rule was used?

43. Definition
If you take the derivative of a derivative, you get a higher-order derivative. The notation is:
Second derivative:

44. Definition
If you take the derivative of a derivative, you get a higher-order derivative. The notation is:
Third derivative:

45. Important Idea
The first derivative represents a rate of change. The second derivative represents the rate of change of the rate of change. In physics, the first derivative is velocity; the second derivative is acceleration.

46. You need your book…

47. Again, b/c these are all over the AP exam…

48. Example
The height above the ground of an object dropped from altitude is:
Find the velocity and acceleration of the object after 10 seconds.

49. Example
The height above the ground of an object dropped from altitude is:
Find the velocity and acceleration of the object after 10 seconds.

50. Lesson Close Without using your notes, what is the product rule?
Without using your notes, what is the quotient rule?

51. Assignment
126/1-43 odd,51,53,86