2.3 Product and Quotient Rules and Higher-Order Derivatives

Transcription

1 Chapter Dierentiation. Prouct an Quotient Rules an Higher-Orer Derivatives Fin the erivative o a unction using the Prouct Rule. Fin the erivative o a unction using the Quotient Rule. Fin the erivative o a trigonometric unction. Fin a higher-orer erivative o a unction. The Prouct Rule In Section., ou learne that the erivative o the sum o two unctions is simpl the sum o their erivatives. The rules or the erivatives o the prouct an quotient o two unctions are not as simple. REMARK A version o the Prouct Rule that some people preer is g g g. The avantage o this orm is that it generalizes easil to proucts o three or more actors. THEOREM.7 Proo Some mathematical proos, such as the proo o the Sum Rule, are straightorwar. Others involve clever steps that ma appear unmotivate to a reaer. This proo involves such a step subtracting an aing the same quantit which is shown in color. g lim 0 Note that lim 0 The Prouct Rule The prouct o two ierentiable unctions an g is itsel ierentiable. Moreover, the erivative o g is the irst unction times the erivative o the secon, plus the secon unction times the erivative o the irst. lim 0 lim 0 g g g g g g g g g g lim g g 0 g lim 0 g g g lim lim g lim g g is continuous. See LarsonCalculus.com or Bruce Ewars s vieo o this proo. lim g 0 because is given to be ierentiable an thereore REMARK The proo o the Prouct Rule or proucts o more than two actors is let as an eercise (see Eercise 7). The Prouct Rule can be etene to cover proucts involving more than two actors. For eample, i, g, an h are ierentiable unctions o, then gh gh gh gh. So, the erivative o sin cos is sin cos cos cos sin sin sin cos cos sin. Copright 0 Cengage Learning. All Rights Reserve. Ma not be copie, scanne, or uplicate, in whole or in part. Due to electronic rights, some thir part content ma be suppresse rom the ebook an/or echapter(s). Eitorial review has eeme that an suppresse content oes not materiall aect the overall learning eperience. Cengage Learning reserves the right to remove aitional content at an time i subsequent rights restrictions require it.

2 . Prouct an Quotient Rules an Higher-Orer Derivatives 9 THE PRODUCT RULE When Leibniz originall wrote a ormula or the Prouct Rule, he was motivate b the epression rom which he subtracte (as being negligible) an obtaine the ierential orm. This erivation resulte in the traitional orm o the Prouct Rule. (Source: The Histor o Mathematics b Davi M. Burton) The erivative o a prouct o two unctions is not (in general) given b the prouct o the erivatives o the two unctions. To see this, tr comparing the prouct o the erivatives o an g 5 with the erivative in Eample. Using the Prouct Rule Fin the erivative o h 5. First Derivative o secon Secon Derivative o irst h 5 5 Appl Prouct Rule In Eample, ou have the option o ining the erivative with or without the Prouct Rule. To in the erivative without the Prouct Rule, ou can write D 5 D 5 5. In the net eample, ou must use the Prouct Rule. Using the Prouct Rule Fin the erivative o sin. sin sin sin cos sin cos sin cos sin Appl Prouct Rule. REMARK In Eample, notice that ou use the Prouct Rule when both actors o the prouct are variable, an ou use the Constant Multiple Rule when one o the actors is a constant. Using the Prouct Rule Fin the erivative o cos sin. cos cos sin sin cos cos sin Prouct Rule Constant Multiple Rule Copright 0 Cengage Learning. All Rights Reserve. Ma not be copie, scanne, or uplicate, in whole or in part. Due to electronic rights, some thir part content ma be suppresse rom the ebook an/or echapter(s). Eitorial review has eeme that an suppresse content oes not materiall aect the overall learning eperience. Cengage Learning reserves the right to remove aitional content at an time i subsequent rights restrictions require it.

3 0 Chapter Dierentiation The Quotient Rule REMARK From the Quotient Rule, ou can see that the erivative o a quotient is not (in general) the quotient o the erivatives. TECHNOLOGY A graphing utilit can be use to compare the graph o a unction with the graph o its erivative. For instance, in Figure., the graph o the unction in Eample appears to have two points that have horizontal tangent lines. What are the values o at these two points? THEOREM. Proo As with the proo o Theorem.7, the ke to this proo is subtracting an aing the same quantit. g lim 0 Deinition o erivative Note that lim g g because g is given to be ierentiable an thereore 0 g g lim 0 gg g g g g lim 0 gg g g g lim lim 0 0 g g lim 0 The Quotient Rule The quotient g o two ierentiable unctions an g is itsel ierentiable at all values o or which g 0. Moreover, the erivative o g is given b the enominator times the erivative o the numerator minus the numerator times the erivative o the enominator, all ivie b the square o the enominator. g g g, g g g g g lim 0 lim 0 gg lim 0 g 0 gg is continuous. See LarsonCalculus.com or Bruce Ewars s vieo o this proo. Using the Quotient Rule g g = ( + ) 7 = 5 + Graphical comparison o a unction an its erivative Figure. 5 Fin the erivative o Appl Quotient Rule. Copright 0 Cengage Learning. All Rights Reserve. Ma not be copie, scanne, or uplicate, in whole or in part. Due to electronic rights, some thir part content ma be suppresse rom the ebook an/or echapter(s). Eitorial review has eeme that an suppresse content oes not materiall aect the overall learning eperience. Cengage Learning reserves the right to remove aitional content at an time i subsequent rights restrictions require it.

4 . Prouct an Quotient Rules an Higher-Orer Derivatives Note the use o parentheses in Eample. A liberal use o parentheses is recommene or all tpes o ierentiation problems. For instance, with the Quotient Rule, it is a goo iea to enclose all actors an erivatives in parentheses, an to pa special attention to the subtraction require in the numerator. When ierentiation rules were introuce in the preceing section, the nee or rewriting beore ierentiating was emphasize. The net eample illustrates this point with the Quotient Rule. Rewriting Beore Dierentiating () = = (, ) The line is tangent to the graph o at the point,. Figure. Fin an equation o the tangent line to the graph o Begin b rewriting the unction. 5 Net, appl the Quotient Rule To in the slope at,, evaluate Write original unction. at,. Multipl numerator an enominator b. Rewrite. Quotient Rule Simpli. 5 Slope o graph at, Then, using the point-slope orm o the equation o a line, ou can etermine that the equation o the tangent line at, is. See Figure.. Not ever quotient nees to be ierentiate b the Quotient Rule. For instance, each quotient in the net eample can be consiere as the prouct o a constant times a unction o. In such cases, it is more convenient to use the Constant Multiple Rule. REMARK To see the beneit o using the Constant Multiple Rule or some quotients, tr using the Quotient Rule to ierentiate the unctions in Eample ou shoul obtain the same results, but with more work. a. b. c. Using the Constant Multiple Rule Original Function Rewrite Dierentiate Simpli Copright 0 Cengage Learning. All Rights Reserve. Ma not be copie, scanne, or uplicate, in whole or in part. Due to electronic rights, some thir part content ma be suppresse rom the ebook an/or echapter(s). Eitorial review has eeme that an suppresse content oes not materiall aect the overall learning eperience. Cengage Learning reserves the right to remove aitional content at an time i subsequent rights restrictions require it.

5 Chapter Dierentiation In Section., the Power Rule was prove onl or the case in which the eponent n is a positive integer greater than. The net eample etens the proo to inclue negative integer eponents. Power Rule: Negative Integer Eponents I n is a negative integer, then there eists a positive integer k such that n k. So, b the Quotient Rule, ou can write n k So, the Power Rule k 0 k k k 0 kk k k k n n. n n n Quotient Rule an Power Rule n k Power Rule is vali or an integer. In Eercise 7 in Section.5, ou are aske to prove the case or which n is an rational number. Derivatives o Trigonometric Functions Knowing the erivatives o the sine an cosine unctions, ou can use the Quotient Rule to in the erivatives o the our remaining trigonometric unctions. THEOREM.9 tan sec sec sec tan Derivatives o Trigonometric Functions cot csc csc csc cot REMARK In the proo o Theorem.9, note the use o the trigonometric ientities an sin cos sec cos. These trigonometric ientities an others are liste in Appeni C an on the ormula cars or this tet. Proo Consiering tan sin cos an appling the Quotient Rule, ou obtain tan sin cos cos cos sin sin cos cos sin cos cos sec. See LarsonCalculus.com or Bruce Ewars s vieo o this proo. Appl Quotient Rule. The proos o the other three parts o the theorem are let as an eercise (see Eercise 7). Copright 0 Cengage Learning. All Rights Reserve. Ma not be copie, scanne, or uplicate, in whole or in part. Due to electronic rights, some thir part content ma be suppresse rom the ebook an/or echapter(s). Eitorial review has eeme that an suppresse content oes not materiall aect the overall learning eperience. Cengage Learning reserves the right to remove aitional content at an time i subsequent rights restrictions require it.

6 . Prouct an Quotient Rules an Higher-Orer Derivatives Dierentiating Trigonometric Functions See LarsonCalculus.com or an interactive version o this tpe o eample. a. b. Function tan sec Derivative sec sec tan sec sec tan REMARK Because o trigonometric ientities, the erivative o a trigonometric unction can take man orms. This presents a challenge when ou are tring to match our answers to those given in the back o the tet. Dierentiate both orms o First orm: Secon orm: cos sin Dierent Forms o a Derivative To show that the two erivatives are equal, ou can write cos sin sin cos cos sin csc cot. cos sin sin sin cos cos sin cos sin csc cot csc cot csc cos sin sin sin sin cos sin csc csc cot. sin cos The summar below shows that much o the work in obtaining a simpliie orm o a erivative occurs ater ierentiating. Note that two characteristics o a simpliie orm are the absence o negative eponents an the combining o like terms. Ater Dierentiating Simpliing Eample 5 5 Eample sin cos cos sin Eample Eample 5 Eample sin sin cos cos sin cos sin Copright 0 Cengage Learning. All Rights Reserve. Ma not be copie, scanne, or uplicate, in whole or in part. Due to electronic rights, some thir part content ma be suppresse rom the ebook an/or echapter(s). Eitorial review has eeme that an suppresse content oes not materiall aect the overall learning eperience. Cengage Learning reserves the right to remove aitional content at an time i subsequent rights restrictions require it.

7 Chapter Dierentiation REMARK The secon erivative o a unction is the erivative o the irst erivative o the unction. Higher-Orer Derivatives Just as ou can obtain a velocit unction b ierentiating a position unction, ou can obtain an acceleration unction b ierentiating a velocit unction. Another wa o looking at this is that ou can obtain an acceleration unction b ierentiating a position unction twice. st vt st at vt st Position unction Velocit unction Acceleration unction The unction at is the secon erivative o st an is enote b st. The secon erivative is an eample o a higher-orer erivative. You can eine erivatives o an positive integer orer. For instance, the thir erivative is the erivative o the secon erivative. Higher-orer erivatives are enote as shown below. First erivative: Secon erivative:, Thir erivative: Fourth erivative: nth erivative:,,, n,,,,, n,,,,, n n,,,,, n n, D D D D D n Fining the Acceleration Due to Gravit The moon s mass is kilograms, an Earth s mass is kilograms.the moon s raius is 77 kilometers, an Earth s raius is 7 kilometers. Because the gravitational orce on the surace o a planet is irectl proportional to its mass an inversel proportional to the square o its raius, the ratio o the gravitational orce on Earth to the gravitational orce on the moon is Because the moon has no atmosphere, a alling object on the moon encounters no air resistance. In 97, astronaut Davi Scott emonstrate that a eather an a hammer all at the same rate on the moon. The position unction or each o these alling objects is st 0.t where st is the height in meters an t is the time in secons, as shown in the igure at the right. What is the ratio o Earth s gravitational orce to the moon s? To in the acceleration, ierentiate the position unction twice. st 0.t st.t st. Position unction Velocit unction Acceleration unction So, the acceleration ue to gravit on the moon is. meters per secon per secon. Because the acceleration ue to gravit on Earth is 9. meters per secon per secon, the ratio o Earth s gravitational orce to the moon s is Earth s gravitational orce 9. Moon s gravitational orce. NASA.0. s s(t) = 0.t + t Copright 0 Cengage Learning. All Rights Reserve. Ma not be copie, scanne, or uplicate, in whole or in part. Due to electronic rights, some thir part content ma be suppresse rom the ebook an/or echapter(s). Eitorial review has eeme that an suppresse content oes not materiall aect the overall learning eperience. Cengage Learning reserves the right to remove aitional content at an time i subsequent rights restrictions require it.

8 . Prouct an Quotient Rules an Higher-Orer Derivatives 5. Eercises See CalcChat.com or tutorial help an worke-out solutions to o-numbere eercises. Using the Prouct Rule In Eercises, use the Prouct Rule to in the erivative o the unction.. g. 5. ht t t. gs ss 5. cos. g sin Using the Quotient Rule In Eercises 7, use the Quotient Rule to in the erivative o the unction g sin. Fining an Evaluating a Derivative In Eercises, in an c Function Value o c Using the Constant Multiple Rule In Eercises 9, complete the table to in the erivative o the unction without using the Quotient Rule Function Rewrite Dierentiate Simpli cos sin 7 5 gt t t 5 9. h 0. t cos t t c 0 c c c c c Fining a Derivative In Eercises 5, in the erivative o the algebraic unction hs s. h c c is a constant c,. c c is a constant c, Fining a Derivative o a Trigonometric Function In Eercises 9 5, in the erivative o the trigonometric unction. 9. t t sin t 0. cos. t cos t. t. tan. cot 5. gt t. h csc t sec sin 7.. cos 9. csc sin 50. sin cos 5. tan 5. sin cos 5. sin cos 5. h 5 sec Fining a Derivative Using Technolog In Eercises 55 5, use a computer algebra sstem to in the erivative o the unction g 5 g g sin 5. sin cos 5 sin sec tan Copright 0 Cengage Learning. All Rights Reserve. Ma not be copie, scanne, or uplicate, in whole or in part. Due to electronic rights, some thir part content ma be suppresse rom the ebook an/or echapter(s). Eitorial review has eeme that an suppresse content oes not materiall aect the overall learning eperience. Cengage Learning reserves the right to remove aitional content at an time i subsequent rights restrictions require it.

9 Chapter Dierentiation Evaluating a Derivative In Eercises 59, evaluate the erivative o the unction at the given point. Use a graphing utilit to veri our result. Function csc 59. csc 0. tan cot. ht sec t t. sin sin cos Point Fining an Equation o a Tangent Line In Eercises, (a) in an equation o the tangent line to the graph o at the given point, (b) use a graphing utilit to graph the unction an its tangent line at the point, an (c) use the erivative eature o a graphing utilit to conirm our results..,., 5. 5, 5.,, 7. tan,. sec,, Famous Curves In Eercises 9 7, in an equation o the tangent line to the graph at the given point. (The graphs in Eercises 9 an 70 are calle Witches o Agnesi. The graphs in Eercises 7 an 7 are calle serpentines.) () = + (, 5 Horizontal Tangent Line In Eercises 7 7, etermine the point(s) at which the graph o the unction has a horizontal tangent line () = (, 5 ( ( () = (, ) +, 5,,,,, (, (, 7 () =, Tangent Lines Fin equations o the tangent lines to the graph o that are parallel to the line. Then graph the unction an the tangent lines. 7. Tangent Lines Fin equations o the tangent lines to the graph o that pass through the point, 5. Then graph the unction an the tangent lines. Eploring a Relationship In Eercises 79 an 0, veri that g, an eplain the relationship between an g Evaluating Derivatives In Eercises an, use the graphs o an g. Let p g an q /g.. (a) Fin p.. (a) Fin p. (b) Fin q. 0 0 (b) Fin q7.. Area The length o a rectangle is given b t 5 an its height is t, where t is time in secons an the imensions are in centimeters. Fin the rate o change o the area with respect to time.. Volume The raius o a right circular cliner is given b t an its height is t, where t is time in secons an the imensions are in inches. Fin the rate o change o the volume with respect to time., sin, 5. Inventor Replenishment The orering an transportation cost C or the components use in manuacturing a prouct is C , where C is measure in thousans o ollars an is the orer size in hunres. Fin the rate o change o C with respect to when (a) 0, (b) 5, an (c) 0. What o these rates o change impl about increasing orer size?. Population Growth A population o 500 bacteria is introuce into a culture an grows in number accoring to the equation Pt 500 g g g 5 t 50 t sin where t is measure in hours. Fin the rate at which the population is growing when t. 0 0 g Copright 0 Cengage Learning. All Rights Reserve. Ma not be copie, scanne, or uplicate, in whole or in part. Due to electronic rights, some thir part content ma be suppresse rom the ebook an/or echapter(s). Eitorial review has eeme that an suppresse content oes not materiall aect the overall learning eperience. Cengage Learning reserves the right to remove aitional content at an time i subsequent rights restrictions require it.

10 . Prouct an Quotient Rules an Higher-Orer Derivatives 7 7. Proo Prove the ollowing ierentiation rules. (a) (b) (c) sec sec tan csc csc cot cot csc. Rate o Change Determine whether there eist an values o in the interval 0, such that the rate o change o sec an the rate o change o g csc are equal. 9. Moeling Data The table shows the health care epenitures h (in billions o ollars) in the Unite States an the population p (in millions) o the Unite States or the ears 00 through 009. The ear is represente b t, with t corresponing to 00. (Source: U.S. Centers or Meicare & Meicai Services an U.S. Census Bureau) Year, t h p (a) Use a graphing utilit to in linear moels or the health care epenitures ht an the population pt. (b) Use a graphing utilit to graph each moel oun in part (a). (c) Fin A htpt, then graph A using a graphing utilit. What oes this unction represent? () Fin an interpret At in the contet o these ata. 90. Satellites When satellites observe Earth, the can scan onl part o Earth s surace. Some satellites have sensors that can measure the angle shown in the igure. Let h represent the satellite s istance rom Earth s surace, an let r represent Earth s raius. Fining a Higher-Orer Derivative In Eercises 99 0, in the given higher-orer erivative. 99., 00., 0., 0., Using Relationships In Eercises 0 0, use the given inormation to in. g h an an g h h g h gh g h WRITING ABOUT CONCEPTS 07. Sketching a Graph Sketch the graph o a ierentiable unction such that 0, < 0 or < <, an > 0 or < <. Eplain how ou oun our answer. 0. Sketching a Graph Sketch the graph o a ierentiable unction such that > 0 an < 0 or all real numbers. Eplain how ou oun our answer. Ientiing Graphs In Eercises 09 an 0, the graphs o,, an are shown on the same set o coorinate aes. Ienti each graph. Eplain our reasoning. To print an enlarge cop o the graph, go to MathGraphs.com r r h θ (a) Show that h rcsc. (b) Fin the rate at which h is changing with respect to when (Assume r 90 miles.) 0. Fining a Secon Derivative secon erivative o the unction. In Eercises 9 9, in the sin 9. sec Sketching Graphs In Eercises, the graph o is shown. Sketch the graphs o an. To print an enlarge cop o the graph, go to MathGraphs.com... Copright 0 Cengage Learning. All Rights Reserve. Ma not be copie, scanne, or uplicate, in whole or in part. Due to electronic rights, some thir part content ma be suppresse rom the ebook an/or echapter(s). Eitorial review has eeme that an suppresse content oes not materiall aect the overall learning eperience. Cengage Learning reserves the right to remove aitional content at an time i subsequent rights restrictions require it.

11 Chapter Dierentiation.. 5. Acceleration The velocit o an object in meters per secon is or 0 t. Fin the velocit an acceleration o the object when t. What can be sai about the spee o the object when the velocit an acceleration have opposite signs?. Acceleration The velocit o an automobile starting rom rest is vt π vt t 00t t 5 π where v is measure in eet per secon. Fin the acceleration at (a) 5 secons, (b) 0 secons, an (c) 0 secons. 7. Stopping Distance A car is traveling at a rate o eet per secon (5 miles per hour) when the brakes are applie. The position unction or the car is st.5t t, where s is measure in eet an t is measure in secons. Use this unction to complete the table, an in the average velocit uring each time interval. π t 0 st vt at. HOW DO YOU SEE IT? The igure shows the graphs o the position, velocit, an acceleration unctions o a particle. 5 7 (a) Cop the graphs o the unctions shown. Ienti each graph. Eplain our reasoning. To print an enlarge cop o the graph, go to MathGraphs.com. (b) On our sketch, ienti when the particle spees up an when it slows own. Eplain our reasoning. t π π π Fining a Pattern In Eercises 9 an 0, evelop a general rule or n given n. Fining a Pattern Consier the unction gh. (a) Use the Prouct Rule to generate rules or ining,, an. (b) Use the results o part (a) to write a general rule or n.. Fining a Pattern Develop a general rule or n, where is a ierentiable unction o. Fining a Pattern In Eercises an, in the erivatives o the unction or n,,, an. Use the results to write a general rule or in terms o n.. n sin. Dierential Equations In Eercises 5, veri that the unction satisies the ierential equation. Function 5., > sin cos sin Dierential Equation True or False? In Eercises 9, etermine whether the statement is true or alse. I it is alse, eplain wh or give an eample that shows it is alse. 9. I g, then g. 0. I then 5, I c an gc are zero an h g, then hc 0.. I is an nth-egree polnomial, then n 0.. The secon erivative represents the rate o change o the irst erivative.. I the velocit o an object is constant, then its acceleration is zero. 5. Absolute Value Fin the erivative o Does 0 eist? (Hint: Rewrite the unction as a piecewise unction an then ierentiate each part.). Think About It Let an g be unctions whose irst an secon erivatives eist on an interval I. Which o the ollowing ormulas is (are) true? (a) g g g g cos n Proo Use the Prouct Rule twice to prove that i, g, an h are ierentiable unctions o, then gh gh gh gh. (b). g g g Copright 0 Cengage Learning. All Rights Reserve. Ma not be copie, scanne, or uplicate, in whole or in part. Due to electronic rights, some thir part content ma be suppresse rom the ebook an/or echapter(s). Eitorial review has eeme that an suppresse content oes not materiall aect the overall learning eperience. Cengage Learning reserves the right to remove aitional content at an time i subsequent rights restrictions require it.

12 Answers to O-Numbere Eercises A Section. (page 5).. 5t t 5. cos sin ) cos sin Function Rewrite Dierentiate Simpli cos sin 7 7 7,., > 0 > 0 5., s s c c 9. tt cos t sin t. t sin t cos tt. sec tan 5. csc t cot t 7. sec tan sec t 9. cos cot 5. sec tan 5. cos sin 55. sin 57. sin 59.. csc cot csc, ht sec tt tan t t,. (a) (b) 5. (a) (b) 5 cos ( 5, 5) , 75. 0, 0,, 77. Tangent lines: 7; (c) + = 7 0 A 0 0 (, 0) () = + (, ) + = 79. g. (a) p (b) q. t 5t cm sec 5. (a) $. thousan00 components (b) $0.7 thousan00 components (c) $.0 thousan00 components The cost ecreases with increasing orer size. 7. Proo 9. (a) ht.t pt.9t (b) h(t).t.9t 0 0 A represents the average health care epenitures per person (in thousans o ollars). 7, () At.t 5.t 79,5 At represents the rate o change o the average health care epenitures per person or the given ear t. 0 p(t) 0 (, ) 7. (a) (b) 0 ( π, ( 9. 0 Copright 0 Cengage Learning. All Rights Reserve. Ma not be copie, scanne, or uplicate, in whole or in part. Due to electronic rights, some thir part content ma be suppresse rom the ebook an/or echapter(s). Eitorial review has eeme that an suppresse content oes not materiall aect the overall learning eperience. Cengage Learning reserves the right to remove aitional content at an time i subsequent rights restrictions require it.

13 A Answers to O-Numbere Eercises cos sin Answers will var. 09. Sample answer:,, cos, sin, sin sin 9. False. g g. True. True 5. ; 0 oes not eist. 7. Proo π π 5. v 7 msec a msec The spee o the object is ecreasing. 7. t 0 st vt at The average velocit on 0, is 57.75, on, is.5, on, is.75, an on, is n nn n... n!. (a) gh gh gh gh gh gh gh gh gh gh gh g h (b) n gh n n!!n! ghn n!!n! ghn... n! n!! gn h g n h. n : cos sin n : cos sin n : cos sin n : cos sin General rule: n cos n n sin Copright 0 Cengage Learning. All Rights Reserve. Ma not be copie, scanne, or uplicate, in whole or in part. Due to electronic rights, some thir part content ma be suppresse rom the ebook an/or echapter(s). Eitorial review has eeme that an suppresse content oes not materiall aect the overall learning eperience. Cengage Learning reserves the right to remove aitional content at an time i subsequent rights restrictions require it.