1. Copyright © Cengage Learning. All rights reserved.2
Differentiation
Copyright © Cengage Learning. All rights reserved.

2. Some Rules for Differentiation
2.2
Some Rules for Differentiation
Copyright © Cengage Learning. All rights reserved.

3. Objectives Find the derivatives of functions using the Constant Rule.
Find the derivatives of functions using the Power Rule.
Find the derivatives of functions using the Constant Multiple Rule.
Find the derivatives of functions using the Sum and Difference Rules.

4. Objectives
Use derivatives to answer questions about real-life situations.

5. The Constant Rule

6. The Constant Rule
You have already found derivatives by the limit process. This process is tedious, even for simple functions, but fortunately there are rules that greatly simplify differentiation.
These rules allow you to calculate derivatives without the direct use of limits.

7. Example 1 – Finding Derivatives of Constant Functionsb. c. d.

8. The Power Rule

9. The Power Rule
The binomial expansion process is used in proving a special case of the Power Rule.

10. The Power Rule
For the Power Rule, the case in which n = 1 is worth remembering as a separate differentiation rule.
That is,

11. The Power Rule
This rule is consistent with the fact that the slope of the line given by y = x is 1. (See Figure 2.13.)
The slope of the line y = x is 1.
Figure 2.13

12. The Power Rule
Before differentiating the function y = 1/x2, you should rewrite 1/x2 as x –2.
Rewriting is the first step in many differentiation problems.
Remember that the derivative of a function f is another function that gives the slope of the graph of f at any point at which f is differentiable. So, you can use the derivative to find slopes, as shown in Example 3.

13. Example 3 – Finding the Slope of a Graph
Find the slopes of the graph of
at x = –2, –1, 0, 1, and 2.
Solution:
Begin by using the Power Rule to find the derivative of f.

14. Example 3 – Solution
cont’d
You can use the derivative to find the slopes of the graph of f, as shown.

15. Example 3 – Solution The graph of f is shown in Figure 2.14. cont’d

16. The Constant Multiple Rule

17. The Constant Multiple Rule
The Constant Multiple Rule is as follows:

18. The Constant Multiple Rule
You may find it helpful to combine the Constant Multiple Rule and the Power Rule into one combined rule.
n is a real number, c is a constant.
Keep in mind that
c is a constant.

19. The Constant Multiple Rule
Parentheses can play an important role in the use of the Constant Multiple Rule and the Power Rule.
In Example 6, be sure you understand the mathematical conventions involving the use of parentheses.

20. Example 6 – Using Parentheses When Differentiatingb. c. d.

21. The Constant Multiple Rule
When differentiating functions involving radicals, you should rewrite the function with rational exponents.
For instance, you should rewrite
and you should rewrite

22. Example 7 – Differentiating Radical Functionsb. c.

23. The Sum and Difference Rules

24. The Sum and Difference Rules
To differentiate y = 3x + 2x3, you would probably write
without questioning your answer. The validity of differentiating a sum or difference of functions term by term is given by the Sum and Difference Rules.

25. The Sum and Difference Rules
The Sum and Difference Rules can be extended to the sum or difference of any finite number of functions.
For instance,
if then

26. Example 10 – Finding an Equation of a Tangent Line
Find an equation of the tangent line to the graph of
at the point
Solution:
The derivative of g(x) is which implies that the slope of the graph at the point is

27. Example 10 – Solution
cont’d
as shown in Figure 2.16.
Figure 2.16

28. Example 10 – Solution
cont’d
Using the point-slope form, you can write the equation of the tangent line at as shown.

29. Application

30. Application
In Example 11, you will use a derivative to find the rate of change of a company’s revenue with respect to time.

31. Example 11 – Modeling Revenue
From 2004 through 2009, the revenue R (in millions of dollars) for McDonald’s can be modeled by
where t represents the year, with t = 4 corresponding to At what rate was McDonald’s revenue changing in 2006?
Solution:
One way to answer this question is to find the derivative of the revenue model with respect to time.

32. Example 11 – Solution
cont’d
In 2006 (at t = 6), the rate of change of the revenue with respect to time is given by

33. Example 11 – Solution
cont’d
Because R is measured in millions of dollars and t is measured in years, it follows that the derivative dR/dt is measured in millions of dollars per year.
So, at the end of 2006, McDonald’s revenue was increasing at a rate of about $1941 million per year, as shown in Figure 2.17.
Figure 2.17