1. Chapter 11
Differentiation

2. Chapter 11: Differentiation
Chapter Objectives
To compute derivatives by using the limit definition.
To develop basic differentiation rules.
To interpret the derivative as an instantaneous rate of change.
To apply the product and quotient rules.
To apply the chain rule.

3. Chapter Outline The Derivative Rules for Differentiation
Chapter 11: Differentiation
Chapter Outline
The Derivative
Rules for Differentiation
The Derivative as a Rate of Change
The Product Rule and the Quotient Rule
The Chain Rule and the Power Rule
11.1)
11.2)
11.3)
11.4)
11.5)

4. 11.1 The Derivative
One of the 1st problems of differential calculus is the tangent line problem
Note that some curves
have no tangent line:
Applications to Business and Economics:
Marginal Cost, Marginal Profit, Minimizing Cost, Maximizing profit, Etc…

7. Therefore the equation of tangent line at (1,1) is:

8. Leibnitz notation
Notation:

9. Let’s construct equation of tangent line at a different point:

10. FYI

11. p q

12. a. For f(x) = x2, it must be continuous for all x.
Foregoing a rigorous proof, the idea is that if a function is smooth enough to have a tangent line, so it must be continuous without jumps etc…
a. For f(x) = x2, it must be continuous for all x.
b. For f(p) =(1/2)p, it is not continuous at p = 0, thus the derivative does not exist at p = 0.
Note that converse is not necessarily true as for f(x)=|x|
I.e. this function is continuous at x = 0, but NOT differentiable

13. Section HW 7 25

14. Section HW 7 25

15. 11.2 Rules for Differentiation
Example:

16. This rule is actually valid for any real power!

17. Differentiate the following functions: Solution: a.
Chapter 11: Differentiation
11.2 Rules for Differentiation
Example 3 – Rewriting Functions in the Form xn
Differentiate the following functions:
Solution: a. b.

18. Example – Using the Power Rule
In Example 2(c), note that before differentiating, 1/x2 was rewritten as x-2.
Rewriting is the first step in many differentiation problems.

20. Differentiate the following functions:
Chapter 11: Differentiation
11.2 Rules for Differentiation
Example 5 – Differentiating Sums and Differences of Functions
Differentiate the following functions:

21. Chapter 11: Differentiation
11.2 Rules for Differentiation
Example 5 – Differentiating Sums and Differences of Functions

22. Find an equation of the tangent line to the curve when x = 1.
Chapter 11: Differentiation
11.2 Rules for Differentiation
Example 7 – Finding an Equation of a Tangent Line
Find an equation of the tangent line to the curve when x = 1.
Solution: The slope equation is
When x = 1,
The equation is

23. Section HW

25. 11.3 The Derivative as a Rate of Change

27. Average velocity is given by Velocity at time t is given by
Chapter 11: Differentiation
Example 1 – Finding Average Velocity and Velocity
Average velocity is given by
Velocity at time t is given by
Suppose the position function of an object moving along a number line is given by s = f(t) = 3t2 + 5, where t is in seconds and s is in meters.
Find the average velocity over the interval [10,
10.1].
b. Find the velocity when t = 10.

28. b. Velocity at time t is given by
Chapter 11: Differentiation
11.3 The Derivative as a Rate of Change
Example 1 – Finding Average Velocity and Velocity
Solution:
a. When t = 10,
b. Velocity at time t is given by
When t = 10, the velocity is
s = f(t) = 3t2 + 5

29. Generalization:

30. Demand and supply curves were discussed in Ch3.
They give dependence between the price/per unit and the quantity demanded by consumers of quantity produced by suppliers

31. Solution: Rate of change of V with respect to r is
Chapter 11: Differentiation
11.3 The Derivative as a Rate of Change
Example 5 – Rate of Change of Volume
A spherical balloon is being filled with air. Find the rate of change of the volume of air in the balloon with respect to its radius. Evaluate this rate of change when the radius is 2 ft.
Solution: Rate of change of V with respect to r is
When r = 2 ft,
This means that when radius is 2, changing it by 1 unit, will change volume by 16π units.

36. In general:

37. Determine the relative and percentage rates of change of
Chapter 11: Differentiation
11.3 The Derivative as a Rate of Change
Example 9 – Relative and Percentage Rates of Change
Determine the relative and percentage rates of change of
when x = 5.
Solution:

38. 11.3 HW

40. 11.4 The Product Rule and the Quotient Rule
Alternative method:
-- seems easier, but for products of transcendental functions (exp, ln, sin/cos etc…) it is essential

41. Application to tangent lines:

42. The Quotient Rule

46. Note that in the demand equation:
p = price/unit
q = number of units

49. 11.4 HW
Differentiate:

50. 11.4 HW

52. Generalizes to:

57. HW

58. HW