1. Chapter 11
Differentiation

2. INTRODUCTORY MATHEMATICAL ANALYSIS
0. Review of Algebra
Applications and More Algebra
Functions and Graphs
Lines, Parabolas, and Systems
Exponential and Logarithmic Functions
Mathematics of Finance
Matrix Algebra
Linear Programming
Introduction to Probability and Statistics

3. INTRODUCTORY MATHEMATICAL ANALYSIS
Additional Topics in Probability
Limits and Continuity
Differentiation
Additional Differentiation Topics
Curve Sketching
Integration
Methods and Applications of Integration
Continuous Random Variables
Multivariable Calculus

4. 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.

5. 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)

6. 11.1 The Derivative Tangent line at a point:
Chapter 11: Differentiation
11.1 The Derivative
Tangent line at a point:
The slope of a curve at P is the slope of the tangent line at P.
The slope of the tangent line at (a, f(a)) is

7. Chapter 11: Differentiation
11.1 The Derivative
Example 1 – Finding the Slope of a Tangent Line
Find the slope of the tangent line to the curve y = f(x) = x2 at the point (1, 1).
Solution: Slope =
The derivative of a function f is the function denoted f’ and defined by

8. Chapter 11: Differentiation
11.1 The Derivative
Example 3 – Finding an Equation of a Tangent Line
If f (x) = 2x2 + 2x + 3, find an equation of the tangent line to the graph of f at (1, 7).
Solution:
Slope 
Equation 

9. a. For f(x) = x2, it must be continuous for all x.
Chapter 11: Differentiation
11.1 The Derivative
Example 5 – A Function with a Vertical Tangent Line
Example 7 – Continuity and Differentiability
Find
Solution:
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.

10. 11.2 Rules for Differentiation
Chapter 11: Differentiation
11.2 Rules for Differentiation
Rules for Differentiation:
RULE 1 Derivative of a Constant:
RULE 2 Derivative of xn:
RULE 3 Constant Factor Rule:
RULE 4 Sum or Difference Rule

11. Chapter 11: Differentiation
11.2 Rules for Differentiation
Example 1 – Derivatives of Constant Functions a.
b. If , then
c. If , then

12. 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.

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

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

15. 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

16. 11.3 The Derivative as a Rate of Change
Chapter 11: Differentiation
11.3 The Derivative as a Rate of Change
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.

17. 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

18. Chapter 11: Differentiation
11.3 The Derivative as a Rate of Change
Example 3 – Finding a Rate of Change
If y = f(x),
then
Find the rate of change of y = x4 with respect to x, and evaluate it when x = 2 and when x = −1.
Solution:
The rate of change is .

19. 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,

20. Applications of Rate of Change to Economics
Chapter 11: Differentiation
11.3 The Derivative as a Rate of Change
Applications of Rate of Change to Economics
Total-cost function is c = f(q).
Marginal cost is defined as .
Total-revenue function is r = f(q).
Marginal revenue is defined as
Relative and Percentage Rates of Change
The relative rate of change of f(x) is
The percentage rate of change of f (x) is

21. If a manufacturer’s average-cost equation is
Chapter 11: Differentiation
11.3 The Derivative as a Rate of Change
Example 7 – Marginal Cost
If a manufacturer’s average-cost equation is
find the marginal-cost function. What is the marginal cost when 50 units are produced?
Solution: The cost is
Marginal cost when q = 50,

22. 11.4 The Product Rule and the Quotient Rule
Chapter 11: Differentiation
11.3 The Derivative as a Rate of Change
Example 9 – Relative and Percentage Rates of Change
11.4 The Product Rule and the Quotient Rule
Determine the relative and percentage rates of change of
when x = 5.
Solution:
The Product Rule

23. Find F’(x). Find y’. Example 1 – Applying the Product Rule
Chapter 11: Differentiation
11.4 The Product and Quotient Rule
Example 1 – Applying the Product Rule
Example 3 – Differentiating a Product of Three Factors
Find F’(x).
Find y’.

24. The Quotient Rule If , find F’(x). Solution:
Chapter 11: Differentiation
11.4 The Product and Quotient Rule
Example 5 – Applying the Quotient Rule
The Quotient Rule
If , find F’(x).
Solution:

25. Differentiate the following functions.
Chapter 11: Differentiation
11.4 The Product and Quotient Rule
Example 7 – Differentiating Quotients without Using the Quotient Rule
Differentiate the following functions.

26. Chapter 11: Differentiation
11.4 The Product and Quotient Rule
Example 9 – Finding Marginal Propensities to Consume and to Save
Consumption Function
If the consumption function is given by determine the marginal propensity to consume and the marginal propensity to save when I = 100.
Solution:

27. 11.5 The Chain Rule and the Power Rule
Chapter 11: Differentiation
11.5 The Chain Rule and the Power Rule
Example 1 – Using the Chain Rule
Chain Rule:
Power Rule:
a. If y = 2u2 − 3u − 2 and u = x2 + 4, find dy/dx.
Solution:

28. b. If y = √w and w = 7 − t3, find dy/dt. Solution:
Chapter 11: Differentiation
11.5 The Chain Rule and the Power Rule
Example 1 – Using the Chain Rule
Example 3 – Using the Power Rule
b. If y = √w and w = 7 − t3, find dy/dt.
Solution:
If y = (x3 − 1)7, find y’.
Solution:

29. If , find dy/dx. Solution: If , find y’. Solution:
Chapter 11: Differentiation
11.5 The Chain Rule and the Power Rule
Example 5 – Using the Power Rule
Example 7 – Differentiating a Product of Powers
If , find dy/dx.
Solution:
If , find y’.
Solution: