Average and Instantaneous Rates of Change: The Derivative, Summaries of Business

Download Average and Instantaneous Rates of Change: The Derivative and more Summaries Business in PDF only on Docsity! 9.3 Average and Instantaneous Rates of Change: The Derivative ! 609 Average Rate of Change Average and Instantaneous Rates of Change: The Derivative ] Application Preview In Chapter 1, “Linear Equations and Functions,” we studied linear revenue functions and defined the marginal revenue for a product as the rate of change of the revenue function. For linear rev- enue functions, this rate is also the slope of the line that is the graph of the revenue function. In this section, we will define marginal revenue as the rate of change of the revenue function, even when the revenue function is not linear. Thus, if an oil company’s revenue (in thousands of dollars) is given by x ! 0 where x is the number of thousands of barrels of oil sold per day, we can find and interpret the marginal revenue when 20,000 barrels are sold (see Example 4). We will discuss the relationship between the marginal revenue at a given point and the slope of the line tangent to the revenue function at that point. We will see how the derivative of the rev- enue function can be used to find both the slope of this tangent line and the marginal revenue. For linear functions, we have seen that the slope of the line measures the average rate of change of the function and can be found from any two points on the line. However, for a function that is not linear, the slope between different pairs of points no longer always gives the same number, but it can be interpreted as an average rate of change. We use this con- nection between average rates of change and slopes for linear functions to define the aver- age rate of change for any function. The average rate of change of a function from to is defined by The figure shows that this average rate is the same as the slope of the segment joining the points (a, f(a)) and (b, f(b)). ! EXAMPLE 1 Total Cost Suppose a company’s total cost in dollars to produce x units of its product is given by Find the average rate of change of total cost for (a) the first 100 units produced (from to ) and (b) the second 100 units produced.x ! 100 x ! 0 C(x) ! 0.01x2 " 25x " 1500 Average rate of change ! f(b) # f(a) b # a x ! bx ! ay ! f(x) R ! 100x " x2, 9.3 OBJECTIVES ! To define and find average rates of change ! To define the derivative as a rate of change ! To use the definition of derivative to find derivatives of functions ! To use derivatives to find slopes of tangents to curves Average Rates of Change x y a b y = f (x) (a, f (a)) (b, f (b)) f (b) – f (a) b – a m = 610 ! Chapter 9 Derivatives Solution (a) The average rate of change of total cost from to units is (b) The average rate of change of total cost from to units is ! EXAMPLE 2 Elderly in the Work Force Figure 9.18 shows the percents of elderly men and of elderly women in the work force in selected census years from 1890 to 2000. For the years from 1950 to 2000, find and inter- pret the average rate of change of the percent of (a) elderly men in the work force and (b) elderly women in the work force. (c) What caused these trends? Solution (a) From 1950 to 2000, the annual average rate of change in the percent of elderly men in the work force is This means that from 1950 to 2000, on average, the percent of elderly men in the work force dropped by 0.456% per year. (b) Similarly, the average rate of change for women is In like manner, this means that from 1950 to 2000, on average, the percent of elderly women in the work force increased by 0.044% each year. (c) In general, from 1950 to 1990, people have been retiring earlier, but the number of women in the work force has increased dramatically. Change in women’s percent Change in years ! 10.0 # 7.8 2000 # 1950 ! 2.2 50 ! 0.044 percent per year Change in men’s percent Change in years ! 18.6 # 41.4 2000 # 1950 ! #22.8 50 ! #0.456 percent per year 0.0% 10.0% 20.0% 30.0% 40.0% 50.0% 60.0% 70.0% 80.0% 1890 1900 1920 1930 1940 1950 1960 1970 1980 1990 2000 Elderly in the Labor Force, 1890-2000 (labor force participation rate; figs. for 1910 not available) 68.3 7.6 63.1 8.3 55.6 54.0 41.8 6.1 41.4 30.5 10.3 24.8 10.0 19.3 8.2 17.6 8.4 18.6 10.07.87.37.3 Men Women ! 6900 # 4100 100 ! 2800 100 ! 28 dollars per unit C(200) # C(100) 200 # 100 ! (0.01(200)2 " 25(200) " 1500) # (4100) 100 x ! 200x ! 100 ! 4100 # 1500 100 ! 2600 100 ! 26 dollars per unit C(100) # C(0) 100 # 0 ! (0.01(100)2 " 25(100) " 1500) # (1500) 100 x ! 100x ! 0 Figure 9.18 Source: Bureau of the Census, U.S. Department of Commerce 9.3 Average and Instantaneous Rates of Change: The Derivative ! 613 Note that in the example above, we could have found the derivative of the function at a particular value of x, say by evaluating the derivative formula at that value: In addition to , the derivative at any point x may be denoted by We can, of course, use variables other than x and y to represent functions and their derivatives. For example, we can represent the derivative of the function defined by 1. Find the average rate of change of over [1, 4]. 2. For the function find (a) (b) (c) (d) In Section 1.6, “Applications of Functions in Business and Economics,” we defined the marginal revenue for a product as the rate of change of the total revenue function for the product. If the total revenue function for a product is not linear, we define the marginal rev- enue for the product as the instantaneous rate of change, or the derivative, of the revenue function. f ¿(2)f ¿(x) ! lim hS0 f(x " h) # f(x) h f(x " h) # f(x) h f(x " h) # f(x) y ! f(x) ! x2 # x " 1, f(x) ! 30 # x # x2 p ! 2q2 # 1 by dp!dq. dy dx , y¿, d dx f(x), Dxy, or Dx f(x) f ¿(x) f ¿(x) ! 8x so f ¿(3) ! 8(3) ! 24 x ! 3,f(x) ! 4x2 Derivative Using the Definition Procedure Example To find the derivative of at any value x: 1. Let h represent the change in x from x to 2. The corresponding change in is 3. Form the difference quotient and simplify. 4. Find to determine the derivative of f(x). f ¿(x),lim hS0 f(x " h) # f(x) h f(x " h) # f(x) h f(x " h) # f(x) y ! f(x) x " h. y ! f(x) Find the derivative of 1. The change in x from x to is h. 2. The change in f(x) is 3. 4. f ¿(x) ! lim hS0 (8x " 4h) ! 8x f ¿(x) ! lim hS0 f(x " h) # f(x) h ! 8x " 4h f(x " h) # f(x) h ! 8xh " 4h2 h ! 8xh " 4h2 ! 4x2 " 8xh " 4h2 # 4x2 ! 4(x2 " 2xh " h2) # 4x2 f(x " h) # f(x) ! 4(x " h)2 # 4x2 x " h f(x) ! 4x2. ! Checkpoint 614 ! Chapter 9 Derivatives Marginal Revenue Suppose that the total revenue function for a product is given by where x is the number of units sold. Then the marginal revenue at x units is provided that the limit exists. Note that the marginal revenue (derivative of the revenue function) can be found by using the steps in the Procedure/Example table on the preceding page. These steps can also be combined, as they are in Example 4. ] EXAMPLE 4 Revenue (Application Preview) Suppose that an oil company’s revenue (in thousands of dollars) is given by the equation where x is the number of thousands of barrels of oil sold each day. (a) Find the function that gives the marginal revenue at any value of x. (b) Find the marginal revenue when 20,000 barrels are sold (that is, at ). Solution (a) The marginal revenue function is the derivative of R(x). Thus, the marginal revenue function is (b) The function found in (a) gives the marginal revenue at any value of x. To find the marginal revenue when 20 units are sold, we evaluate Hence the marginal revenue at is $60,000 per thousand barrels of oil. Because the marginal revenue is used to approximate the revenue from the sale of one additional unit, we interpret to mean that the expected revenue from the sale of the next thousand barrels (after 20,000) will be approximately $60,000. [Note: The actual revenue from this sale is (thousand dollars).] As mentioned earlier, the rate of change of revenue (the marginal revenue) for a linear rev- enue function is given by the slope of the line. In fact, the slope of the revenue curve gives us the marginal revenue even if the revenue function is not linear. We will show that the slope of the graph of a function at any point is the same as the derivative at that point. In order to show this, we must define the slope of a curve at a point on the curve. We will define the slope of a curve at a point as the slope of the line tangent to the curve at the point. R(21) # R(20) ! 1659 # 1600 ! 59 R¿(20) ! 60 x ! 20 R¿(20) ! 100 # 2(20) ! 60 R¿(20). MR ! R¿(x) ! 100 # 2x. ! lim hS0 100h # 2xh # h2 h ! lim hS0 (100 # 2x # h) ! 100 # 2x ! lim hS0 100x " 100h # (x2 " 2xh " h2) # 100x " x2 h ! lim hS0 3100(x " h) # (x " h)2 4 # (100x # x2) h R¿(x) ! lim hS0 R(x " h) # R(x) h x ! 20 R ! R(x) ! 100x # x2, x & 0 MR ! R¿(x) ! lim hS0 R(x " h) # R(x) h R ! R(x), Tangent to a Curve 9.3 Average and Instantaneous Rates of Change: The Derivative ! 615 In geometry, a tangent to a circle is defined as a line that has one point in common with the circle. (See Figure 9.20(a).) This definition does not apply to all curves, as Figure 9.20(b) shows. Many lines can be drawn through the point A that touch the curve only at A. One of the lines, line l, looks like it is tangent to the curve. We can use secant lines (lines that intersect the curve at two points) to determine the tangent to a curve at a point. In Figure 9.21, we have a set of secant lines and that pass through a point A on the curve and points and on the curve near A. (For points and secant lines to the left of point A, there would be a similar figure and discussion.) The line l represents the tangent line to the curve at point A. We can get a secant line as close as we wish to the tangent line l by choosing a “second point” Q suffi- ciently close to point A. As we choose points on the curve closer and closer to A (from both sides of A), the limiting position of the secant lines that pass through A is the tangent line to the curve at point A, and the slopes of those secant lines approach the slope of the tangent line at A. Thus we can find the slope of the tangent line by finding the slope of a secant line and tak- ing the limit of this slope as the “second point” Q approaches A. To find the slope of the tangent to the graph of at we first draw a secant line from point A to a second point on the curve (see Figure 9.22).Q(x1 " h, f(x1 " h)) A(x1, f(x1)),y ! f(x) Q4Q1, Q2, Q3, s4s1, s2, s3, x y A l (b) A (a)Figure 9.20 Figure 9.21 Figure 9.22 x y A(x1, f (x1)) Q(x1 + h, f (x1 + h)) h f (x1 + h) ! f (x1) y = f (x) x y A Q4 Q3 Q2 Q1 l s4 s3 s2 s1 The slope of this secant line is As Q approaches A, we see that the difference between the x-coordinates of these two points decreases, so h approaches 0. Thus the slope of the tangent is given by the following. mAQ ! f(x1 " h) # f(x1) h 618 ! Chapter 9 Derivatives In addition, we know that the slope of the tangent to f(x) at is defined by Hence we could also estimate —that is, the slope of the tangent at —by evaluating ! EXAMPLE 7 Approximating the Slope of the Tangent Line (a) Let Use and two values of h to make estimates of the slope of the tangent to f(x) at on opposite sides of (b) Use the following table of values of x and g(x) to estimate x 1 1.9 2.7 2.9 2.999 3 3.002 3.1 4 5 g(x) 1.6 4.3 11.4 10.8 10.513 10.5 10.474 10.18 6 #5 Solution The table feature of a graphing utility can facilitate the following calculations. (a) We can use and as follows: (b) We use the given table and measure the slope between (3, 10.5) and another point that is nearby (the closer, the better). Using (2.999, 10.513), we obtain Most graphing calculators have a feature called the numerical derivative (usually denoted by nDer or nDeriv) that can approximate the derivative of a function at a point. On most calculators this feature uses a calculation similar to our method in part (a) of Example 7 and produces the same estimate. The numerical derivative of with respect to x at can be found as follows on many graphing calculators: " So far we have talked about how the derivative is defined, what it represents, and how to find it. However, there are functions for which derivatives do not exist at every value of x. Figure 9.24 shows some common cases where does not exist but where exists for all other values of x. These cases occur where there is a discontinuity, a corner, or a vertical tangent line. f ¿(x)f ¿(c) nDeriv(2x3 # 6x2 " 2x # 5, x, 3) ! 20 x ! 3 f(x) ! 2x3 # 6x2 " 2x # 5 g¿(3) " y2 # y1 x2 # x1 ! 10.5 # 10.513 3 # 2.999 ! #0.013 0.001 ! #13 ! f (2.9999) # f(3) #0.0001 ! 19.9988 " 20 With h ! #0.0001: f ¿(3) " f(3 " (#0.0001)) # f(3) #0.0001 ! f(3.0001) # f (3) 0.0001 ! 20.0012 " 20 With h ! 0.0001: f ¿(3) " f(3 " 0.0001) # f(3) 0.0001 h ! #0.0001h ! 0.0001 g¿(3). x ! 3.x ! 3 f(a " h) # f(a) h f(x) ! 2x3 # 6x2 " 2x # 5. f(a " h) # f(a) h when h " 0 and h ' 0 x ! af ¿(a) f ¿(a) ! lim hS0 f(a " h) # f(a) h x ! a Differentiability and Continuity Calculator Note Differentiability Implies Continuity 9.3 Average and Instantaneous Rates of Change: The Derivative ! 619 From Figure 9.24 we see that a function may be continuous at even though does not exist. Thus continuity does not imply differentiability at a point. However, differ- entiability does imply continuity. If a function f is differentiable at then f is continuous at ! EXAMPLE 8 Water Usage Costs The monthly charge for water in a small town is given by (a) Is this function continuous at (b) Is this function differentiable at Solution (a) We must check the three properties for continuity. 1. 2. 3. Thus f(x) is continuous at (b) Because the function is defined differently on either side of we need to test to see whether exists by evaluating both (i) and (ii) and determining whether they are equal. (i) ! lim hS0# 0 ! 0 lim hS0# f(20 " h) # f(20) h ! lim hS0# 18 # 18 h lim hS0" f(20 " h) # f(20) h lim hS0# f(20 " h) # f(20) h f ¿(20) x ! 20, x ! 20. lim xS20 f(x) ! f(20) lim xS20" f(x) ! lim xS20" (0.1x " 16) ! 18 lim xS20# f(x) ! lim xS20# 18 ! 18 f(x) ! 18 for x $ 20 so f (20) ! 18 x ! 20? x ! 20? y ! f (x) ! b18 if 0 $ x $ 20 0.1x " 16 if x ( 20 x ! c.x ! c, f ¿(c)x ! c x y c Discontinuity (a) Not differentiable at x = c x y c Corner (b) Not differentiable at x = c Vertical tangent c x y (c) Not differentiable at x = c x y c (d) Not differentiable at x = c Vertical tangent Figure 9.24 r1 lim xS20 f(x) ! 18 620 ! Chapter 9 Derivatives (ii) Because these limits are not equal, the derivative does not exist. 3. Which of the following are given by (a) The slope of the tangent when (b) The y-coordinate of the point where (c) The instantaneous rate of change of f(x) at (d) The marginal revenue at if f(x) is the revenue function 4. Must a graph that has no discontinuity, corner, or cusp at be differentiable at We can use a graphing calculator to explore the relationship between secant lines and tangent lines. For example, if the point (a, b) lies on the graph of then the equa- tion of the secant line to from (1, 1) to (a, b) has the equation Figure 9.25 illustrates the secant lines for three different choices for the point (a, b). y # 1 ! b # 1 a # 1 (x # 1), or y ! b # 1 a # 1 (x # 1) " 1 y ! x2 y ! x2, x ! c? x ! c x ! c, x ! c x ! c x ! c f ¿(c)? f ¿(20) ! lim hS0" 0.1 ! 0.1 ! lim hS0" 0.1h h lim hS0" f(20 " h) # f(20) h ! lim hS0" 30.1(20 " h) " 16 4 # 18 h ! Checkpoint Calculator Note -6 6 -5 25 (a) -6 6 -5 25 (b) -6 6 -5 25 (c) Figure 9.25 We see that as the point (a, b) moves closer to (1, 1), the secant line looks more like the tangent line to at (1, 1). Furthermore, (a, b) approaches (1, 1) as and the slope of the secant approaches the following limit. This limit, 2, is the slope of the tangent line at (1, 1). That is, the derivative of at (1, 1) is 2. [Note that a graphing utility’s calculation of the numerical derivative of with respect to x at gives ] "f ¿(1) ! 2.x ! 1 f(x) ! x2 y ! x2 lim aS1 b # 1 a # 1 ! lim aS1 a2 # 1 a # 1 ! lim aS1 (a " 1) ! 2 a S 1,y ! x2 9.3 Average and Instantaneous Rates of Change: The Derivative ! 623 31. (a) Over what interval(s) (a) through (d ) is the rate of change of f(x) positive? (b) Over what interval(s) (a) through (d ) is the rate of change of f(x) negative? (c) At what point(s) A through E is the rate of change of f(x) equal to zero? 32. (a) At what point(s) A through E does the rate of change of f(x) change from positive to negative? (b) At what point(s) A through E does the rate of change of f(x) change from negative to positive? 33. Given the graph of in Figure 9.27, determine for which x-values A, B, C, D, or E the function is (a) continuous. (b) differentiable. 34. Given the graph of in Figure 9.27, determine for which x-values F, G, H, I, or J the function is (a) continuous. (b) differentiable. y ! f(x) y ! f (x) APPLICATIONS 39. Total cost Suppose total cost in dollars from the pro- duction of x printers is given by Find the average rate of change of total cost when pro- duction changes (a) from 100 to 300 printers. (b) from 300 to 600 printers. (c) Interpret the results from parts (a) and (b). 40. Average velocity If an object is thrown upward at from a height of 20 feet, its height S after x sec- onds is given by What is the average velocity in the (a) first 2 seconds after it is thrown? (b) next 2 seconds? 41. Demand If the demand for a product is given by what is the average rate of change of demand when p increases from (a) 1 to 25? (b) 25 to 100? 42. Revenue If the total revenue function for a blender is where x is the number of units sold, what is the aver- age rate of change in revenue R(x) as x increases from 10 to 20 units? 43. Total cost Suppose the figure shows the total cost graph for a company. Arrange the average rates of change of total cost from A to B, B to C, and A to C from smallest to greatest, and explain your choice. R(x) ! 36x # 0.01x2 D(p) ! 10001p # 1 S(x) ! 20 " 64x # 16x2 64 ft/s C(x) ! 0.0001x3 " 0.005x2 " 28x " 3000 x y A B C D E a b c d Figure 9.26 Figure 9.27 x y A C D E G H I JB F y = f (x) In Problems 35–38, (a) find the slope of the tangent to the graph of f(x) at any point, (b) find the slope of the tangent at the given x-value, (c) write the equation of the line tangent to the graph of f(x) at the given point, and (d) graph both f(x) and its tangent line (use a graphing utility if one is available). 35. (a) 36. (a) (b) (b) (c) (2, 6) (c) 37. (a) 38. (a) (b) (b) (c) (1, 4) (c) (#1, #3) x ! #1x ! 1 f(x) ! 5x3 " 2f(x) ! x3 " 3 (#1, #2) x ! #1x ! 2 f(x) ! x2 " 3xf(x) ! x2 " x 20 40 60 80 100 10 20 30 40 50 x C(x) A B C Thousands of Units T ho us an ds o f d ol la rs 624 ! Chapter 9 Derivatives 44. Students per computer The following figure shows the number of students per computer in U.S. public schools for the school years that ended in 1984 through 2002. (a) Use the figure to find the average rate of change in the number of students per computer from 1990 to 2000. Interpret your result. (b) From the figure, determine for what two consecu- tive school years the average rate of change of the number of students per computer is closest to zero. 45. Marginal revenue Say the revenue function for a stereo system is where x denotes the number of units sold. (a) What is the function that gives marginal revenue? (b) What is the marginal revenue if 50 units are sold, and what does it mean? (c) What is the marginal revenue if 200 units are sold, and what does it mean? (d) What is the marginal revenue if 150 units are sold, and what does it mean? (e) As the number of units sold passes through 150, what happens to revenue? 46. Marginal revenue Suppose the total revenue function for a blender is where x is the number of units sold. (a) What function gives the marginal revenue? (b) What is the marginal revenue when 600 units are sold, and what does it mean? (c) What is the marginal revenue when 2000 units are sold, and what does it mean? (d) What is the marginal revenue when 1800 units are sold, and what does it mean? 47. Labor force and output The monthly output at the Olek Carpet Mill is Q(x) ! 15,000 " 2x2 units, (40 $ x $ 60) R(x) ! 36x # 0.01x2 dollars R(x) ! 300x # x2 dollars where x is the number of workers employed at the mill. If there are currently 50 workers, find the instantaneous rate of change of monthly output with respect to the number of workers. That is, find 48. Consumer expenditure Suppose that the demand for x units of a product is where p dollars is the price per unit. Then the consumer expenditure for the product is What is the instantaneous rate of change of consumer expenditure with respect to price at (a) any price p? (b) (c) In Problems 49–52, find derivatives with the numerical derivative feature of a graphing utility. 49. Profit Suppose that the profit function for the monthly sales of a car by a dealership is where x is the number of cars sold. What is the instan- taneous rate of change of profit when (a) 200 cars are sold? Explain its meaning. (b) 300 cars are sold? Explain its meaning. 50. Profit If the total revenue function for a toy is and the total cost function is what is the instantaneous rate of change of profit if 10 units are produced and sold? Explain its meaning. 51. Heat index The highest recorded temperature in the state of Alaska was and occurred on June 27, 1915, at Fort Yukon. The heat index is the apparent temperature of the air at a given temperature and humidity level. If x denotes the relative humidity (in percent), then the heat index (in degrees Fahrenheit) for an air temperature of can be approximated by the function (a) At what rate is the heat index changing when the humidity is 50%? (b) Write a sentence that explains the meaning of your answer in part (a). 52. Receptivity In learning theory, receptivity is defined as the ability of students to understand a complex concept. Receptivity is highest when the topic is introduced and tends to decrease as time passes in a lecture. Suppose f(x) ! 0.009x2 " 0.139x " 91.875 100°F 100°F C(x) ! 100 " 0.2x2 " x R(x) ! 2x P(x) ! 500x # x2 # 100 p ! 20?p ! 5? ! 10,000p # 100p2 E(p) ! px ! p(10,000 # 100p) x ! 10,000 # 100p Q ¿(50). Students Per Computer in U.S. Public Schools 0 50 100 150 ’8 3– ’8 4 ’8 4– ’8 5 ’8 5– ’8 6 ’8 6– ’8 7 ’8 7 –’ 88 ’8 8 –’ 89 ’8 9– ’9 0 ’9 0– ’9 1 ’9 1– ’9 2 ’9 2– ’9 3 ’9 3– ’9 4 ’9 4– ’9 5 ’9 5– ’9 6 ’9 6– ’9 7 ’9 7– ’9 8 ’9 8– ’9 9 ’9 9– ’0 0 20 00 –’ 01 20 01 –’ 02 12 5. 0 75 .0 50 .0 37 .0 32 .0 25 .0 22 .0 20 .0 18 .0 16 .0 14 .0 10 .5 10 .0 7. 8 6. 1 5. 7 5. 4 5. 0 4. 9 Source: Quality Education Data, Inc., Denver, Co. Reprinted by permission. 9.4 Derivative Formulas ! 625 that the receptivity of a group of students in a mathe- matics class is given by where t is minutes after the lecture begins. (a) At what rate is receptivity changing 10 minutes after the lecture begins? (b) Write a sentence that explains the meaning of your answer in part (a). 53. Marginal revenue Suppose the graph shows a manu- facturer’s total revenue, in thousands of dollars, from the sale of x cellular telephones to dealers. (a) Is the marginal revenue greater at 300 cell phones or at 700? Explain. (b) Use part (a) to decide whether the sale of the 301st cell phone or the 701st brings in more revenue. Explain. g(t) ! #0.2t2 " 3.1t " 32 54. Social Security beneficiaries The graph shows a model for the number of millions of Social Security beneficiaries (actual to 2000 and projected beyond 2000). The model was developed with data from the 2000 Social Security Trustees Report. (a) Was the instantaneous rate of change of the num- ber of beneficiaries with respect to the year greater in 1960 or in 1980? Justify your answer. (b) Is the instantaneous rate of change of the number of beneficiaries projected to be greater in 2000 or in 2030? Justify your answer. 200 400 600 800 1000 20 40 60 80 x y R(x) 1970 1990 2010 2030 40 80 x y Year M ill io ns o f B en ef ic ia ri es Derivative Formulas ] Application Preview For more than 30 years, U.S. total personal income has experienced steady growth. With Bureau of Economic Analysis, U.S. Department of Commerce data for selected years from 1975 to 2002, U.S. total personal income I, in billions of current dollars, can be modeled by I ! I(t ) ! 5.33t2 # 81.5t # 822 where t is the number of years past 1970. We can find the rate of growth of total U.S. personal income in 2007 by using the derivative I (t) of the total personal income function. (See Example 8.) As we discussed in the previous section, the derivative of a function can be used to find the rate of change of the function. In this section we will develop formulas that will make it easier to find certain derivatives. We can use the definition of derivative to show the following: If f (x) ! x5, then f ¿(x) ! 5x4. If f(x) ! x4, then f ¿(x) ! 4x3. If f(x) ! x3, then f ¿(x) ! 3x2. If f(x) ! x2, then f ¿(x) ! 2x. $ 9.4 OBJECTIVES ! To find derivatives of powers of x ! To find derivatives of constant functions ! To find derivatives of functions involving constant coefficients ! To find derivatives of sums and differences of functions Derivative of f(x) ! xn