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

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

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.

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

The Constant Rule

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.

Example 1 – Finding Derivatives of Constant Functions b. c. d.

The Power Rule

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

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

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

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.

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.

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

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

The Constant Multiple Rule

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

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.

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.

Example 6 – Using Parentheses When Differentiating b. c. d.

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

Example 7 – Differentiating Radical Functions b. c.

The Sum and Difference Rules

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.

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

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

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

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

Application

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

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 2004. 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.

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

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