You may recall from your review of slopes in the section on linear functions that the slope has important business implications. Slope tells you the sensitivity of the y variable to changes in the x variable. This sensitivity is expressed in the value of the slope and is sometimes called the "rate of change" because it measures the rate of change in y as a result of a change in x. As the absolute value of the slope increases, so does the sensitivity of y to changes in x. Nonlinear functions have slopes that change; in fact, the slope may be different at every point along the curve.

Slope also tells you the "nature of change" in y as a result of a change in x. This nature of change is expressed in the sign of the slope. A slope may be positive, negative, or zero. Consider the graph of the parabolic function below.

Three points are labeled along this curve: A, B, and C. The slope is different at each of these points. At point A, the slope is zero. This point is the parabola's vertex—the maximum or minimum point of the function. At point B, the slope is negative. At point C, the slope is positive.

Because slope has important business implications, managers are interested in measuring its value. The difficulty arises with nonlinear functions in that the slope changes at every point along the curve. To find the slope of a curve, you must select a point along the curve, draw a line—either through two points on the curve which are very close to the selected point or tangent to the curve at the selected point—and then measure the slope of the line drawn.

If you draw a line through two points that are close to one another on a curve, that line is called a secant line. The slope of a secant line is also known as the "average rate of change." The slope of the secant line approximates the slope of the curve at any point between the two points on the curve. When the two points chosen are closer together, the slope of the secant line becomes closer to the actual slope of the curve at the point of interest.

A tangent line is a line that touches a nonlinear function at only one point, as shown in the right side of the figure above. The tangent line's slope is called the "instantaneous rate of change" because it is a precise measurement of the slope of the curve at that particular point. Finding the equation of the tangent line at any point of a curve can be tricky. Using the first derivative of the nonlinear function, however, you can quickly find the slope of the tangent line at any point on the curve without having to find the equation of the tangent line.

Secant Lines

The slope of a secant line approximates the slope of a nonlinear function between two points. Because a secant line is straight, its slope can be determined using the rise-over-run formula, which is discussed in the Linear Functions portion of the Algebra section of this course.

The slope of a secant line can be used to find the average rate of change of sales revenues, price levels, or any other value represented by a nonlinear function. For example, suppose that the company for which you work produces nuts and bolts. The nuts and bolts are produced using the same manufacturing equipment. The company needs to produce both nuts and bolts, but there is a tradeoff that must be made regarding the use of the equipment—when the company produces more nuts, the manufacturing capacity for bolts is reduced. The relationship of the production of nuts and bolts is described by the following production function.

In this function, f(x) is the number (in millions) of nuts produced and x is the number (in millions) of bolts produced.

The company wants to know more about the production tradeoffs, or average rate of change, between the number of nuts and bolts that can be produced at a rate of 5 million nuts, which corresponds to 2 million bolts produced. You can calculate average rate of change by identifying the slope of a secant line that passes through the point (2, 5) and another point. For the best approximation, you should choose a point on the function that is nearby, such as (2.01, 4.9599). The secant line that passes through the function at these two points is shown below.

You can find the slope of the secant line by calculating the change in y over the change in x between the points (2, 5) and (2.01, 4.9599). As you can see in the graph above, the y-axis is represented as f(x).

You can report to the company that the average rate of change in production is 4.01 million fewer nuts per 1 million more bolts. To interpret this rate of change, recall the slope formula (rise/run). In this area of the curve, for every one million more bolts produced (run), production of nuts will have to be cut by 4.01 million (rise).

First Derivatives

The slope of a secant line only estimates rate of change. It has already been mentioned that this estimation is more exact when you use a secant line that is between two close points on the curve. As you choose these points closer and closer together, the slope approaches an exact value. The line passing through this single point is the tangent line, and its slope, which measures the slope of the function at a single point, is the instantaneous rate of change.

The equation that describes the exact slope at this single point is called the function's first derivative. First derivatives are commonly used to identify the slope of a nonlinear function at a particular point and to identify any maximum and minimum points of a function. The first derivative is often written f'(a), which is read as "f prime of a." A table listing the fundamental rules of derivatives may be viewed by opening the link in the right margin.

The steps for finding a function's first derivative are explained and illustrated in the following animation.

Derivatives View animation

Many of the functions you will encounter will be polynomial expressions such as f(x) = ax² + bx + c. This course will only consider derivatives of polynomial functions such as these. For help finding derivatives of other types of functions, consult a calculator or a business math text.

Practice finding the first derivative of the polynomial expressions below. To find a function's first derivative, multiply the constant of each term by the exponent, then subtract one from each exponent.

Consider a slightly more complex example; find the derivative of the following function.

Now that you have reviewed some of the basics of derivatives, consider again the company that produces nuts and bolts. The company originally wanted to determine the average rate of change of production between producing nut and producing bolts at the level of production where 2 million bolts and 5 million nuts are being produced. Your original calculation using a secant line was an approximation of the rate of change.

Now the company would like a precise measurement of the the instantaneous rate of change, or the slope of the line tangent to the point (2, 5).

To determine this rate, you need to find the first derivative of the company's production function, f(x) = –x² + 9.

Now that you have the first derivative, you can use it to find the slope of the line tangent to the point (2, 5). To do this, insert the x value, 2, into the derivative equation and solve.

The slope, or instantaneous rate of change, of the production function at the point (2, 5) is –4. This means that in order to produce 1 million more bolts, the company will have to produce 4 million fewer nuts.

Compare the difference between using the slope of a secant line (the average rate of change) and the slope of a tangent line (the instantaneous rate of change). The slopes are -4.01 million and -4 million, respectively. This demonstrates how the average rate of change can closely approximate the instantaneous rate of change when the points chosen are very close together.

It is important to know whether the slope of a function at a particular point is positive, negative, or zero. The sign of the slope indicates to managers the nature of the change in the dependent variable for a given change in the independent variable. The sign of the slope can also be easily understood using the first derivative.

• If f '(x) < 0 ; then f(x) is decreasing.
• If f '(x) = 0 ; then f(x) is at a relative maximum or minimum.
• If f '(x) > 0 ; then f(x) is increasing.

Extrema

Businesses often want to identify a nonlinear function's maximum and minimum points, which are also known as extrema. The maximum point on a production function will tell a manager the point at which marginal production levels are at their highest. The minimum point on a cost function will tell managers where marginal unit costs are at their lowest. Extrema occur at the point at which slope is zero. Therefore, maximum and minimum points can be found by taking the first derivative of the function, setting it equal to zero, and solving for x.

This use of derivatives has many important practical applications, including identifying a project team's maximum unit output of work. Consider a department that is assembling a project team to develop a new web site. The company has many projects going on concurrently and, although the web site development project is important, the company would like to staff it optimally so that the project is completed in the fastest manner possible and with the "right" number of people.

The manager of this department hires you to determine the level of staffing at which each team member on the project will have the greatest level of output in terms of the tasks completed per week. You may have experienced this scenario on a project team for a job or school project in the past. If the team has too few people, team members are spread too thin across tasks and seldom get tasks completed. A team with too many people may result in additional time required in meetings to coordinate the activities of all the team members, resulting in fewer tasks being completed by each individual team member.

For this particular team, the function of productivity of an individual team member is stated as f(x) = –x² + 20x – 80.

To begin to determine the optimal number of people for the team, find the first derivative of the productivity function.

When the first derivative is positive, the individual output continues to increase as people are added to the team. When the first derivative is negative, the individual output decreases as people are added to the team. Individual output is maximized at the point at which the individual output can no longer increase and has not yet begun to decrease. This maximum point occurs at the point at which the derivative is zero.

You find that the department should assign 10 people to the development team in order to maximize the individual output. To determine the individual output when 10 people are assigned to the team, insert the quantity 10 into the original function.

The number of tasks completed when individual productivity is maximized is 20 tasks per week. This occurs when 10 people are assigned to the project team.

Second Derivatives

The second derivative can be written as f "(x), which can be expressed verbally as "f double prime of x." While the first derivative measures the rate of change of a function, the second derivative measures whether this rate of change is increasing or decreasing.

To find the second derivative, simply take the derivative of the first derivative. Look at the following example of a function and its first and second derivative.

1. Consider the graph below: Is the slope of the graph at point B positive, negative, or zero?

Solution 1

2. Which point on the following graph has a slope of zero?

Solution 2

3. The two points of a secant line should be _____ to ensure a good approximation of the slope. [fill in blank]

Solution 3

4. Find the first derivative of the following function:

f(x) = 4x³ + 3x² – 8

Solution 4

5. Find the slope of the following function at x = 2:

f(x) = 7x² – 9x + 3

Solution 5

6. If f(x) = x² – 4x + 8 , find the average rate of change between the points x = 2 and x = 2.1.

Solution 6

7. The amount of output a company can produce is dependent on the number of workers. A company's production function is defined as f(x) = –x² + 40x; where x is the number of workers employed, and f(x) is the number of units produced by the company. The company currently has six employees; find the average rate of change (slope of the secant line) of output of adding one more employee.

Solution 7

8. A company produces both color printers and black-and-white printers. Because the materials used to produce the two printers are mostly the same, the company faces a trade off between the number of color printers produced and the number of black-and-white printers produced. The function that models this relationship f(x) = –.25x² + 36; where x is the number of color printers, and f(x) is the number of black-and-white printers.

If the company is currently producing 10 color printers, what is the instantaneous rate of change (slope of the tangent) at this point?

Solution 8

9. A computer company sells a particular desktop model for $900. The company has a cost function of f(x) = 3x² + 36x + 6,000, where x is the number of units produced. To maximize profit, how many desktop computers should be produced, and how much will the company's profit be? (Note: Profit = Revenue – Cost)

Solution 9