Product and Quotient Rules and Higher-Order Derivatives. The Product Rule

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1 330_003.q 11/3/0 :3 PM Page 119 SECTION.3 Prouct an Quotient Rules an Higher-Orer Derivatives 119 Section.3 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. NOTE 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 involving three or more actors. g lim 0 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. lim 0 g lim g g 0 g lim 0 g lim 0 g g g g g g g g lim lim g lim g g 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. g g g lim g 0 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) Note that lim because is given to be ierentiable an thereore 0 is continuous. 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. For instance, the erivative o sin cos is sin cos cos cos sin sin sin cos cos sin.

2 330_003.q 11/3/0 :3 PM Page CHAPTER Dierentiation 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 3 an g 5 with the erivative in Eample 1. EXAMPLE 1 Using the Prouct Rule Fin the erivative o h 3 5. Derivative Derivative First o secon Secon o irst h Appl Prouct Rule In Eample 1, 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 3 5 D In the net eample, ou must use the Prouct Rule. EXAMPLE Using the Prouct Rule Fin the erivative o 3 sin. 3 sin 3 sin sin 3 3 cos sin 3 cos sin 3 cos sin Appl Prouct Rule. EXAMPLE 3 Using the Prouct Rule Fin the erivative o cos sin. NOTE In Eample 3, 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. sin Prouct Rule sin cos cos Constant Multiple Rule cos cos sin

3 330_003.q 11/3/0 :3 PM Page 11 SECTION.3 Prouct an Quotient Rules an Higher-Orer Derivatives 11 The Quotient Rule THEOREM. 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 0 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? = ( + 1) 7 = Graphical comparison o a unction an its erivative Figure. Proo As with the proo o Theorem.7, the ke to this proo is subtracting an aing the same quantit. g lim 0 Note that lim is continuous. EXAMPLE 0 g g lim 0 gg g lim 0 g g g g Deinition o erivative g g g g lim 0 gg g g g lim lim 0 0 g g because g is given to be ierentiable an thereore Using the Quotient Rule 5 Fin the erivative o g lim 0 lim 0 gg lim 0 gg g g Appl Quotient Rule.

4 330_003.q 11/3/0 :3 PM Page 1 1 CHAPTER Dierentiation 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. EXAMPLE 5 Rewriting Beore Dierentiating 3 1 () = = 1 ( 1, 1) The line 1 is tangent to the graph o at the point 1, 1. Figure.3 Fin an equation o the tangent line to the graph o Begin b rewriting the unction To in the slope at 1, 1, evaluate Write original unction. at 1, 1. Multipl numerator an enominator b. Rewrite. Quotient Rule Simpli Slope o graph at 1, 1 Then, using the point-slope orm o the equation o a line, ou can etermine that the equation o the tangent line at 1, 1 is 1. See Figure.3. Not ever quotient nees to be ierentiate b the Quotient Rule. For eample, 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. EXAMPLE Using the Constant Multiple Rule NOTE 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. Original Function Rewrite Dierentiate Simpli

5 330_003.q 11/3/0 :3 PM Page 13 SECTION.3 Prouct an Quotient Rules an Higher-Orer Derivatives 13 In Section., the Power Rule was prove onl or the case where the eponent n is a positive integer greater than 1. The net eample etens the proo to inclue negative integer eponents. EXAMPLE 7 Proo o the Power Rule (Negative Integer Eponents) I n is a negative integer, there eists a positive integer k such that n k. So, b the Quotient Rule, ou can write n 1 k So, the Power Rule k 0 1k k1 k 0 kk1 k k k1 n n1. D n n n1 Quotient Rule an Power Rule Power Rule is vali or an integer. In Eercise 75 in Section.5, ou are aske to prove the case or which n is an rational number. Derivatives o Trigonometric Functions n k 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 Proo Consiering tan sin cos an appling the Quotient Rule, ou obtain cos cos sin sin tan cos cos sin cos 1 cos sec. Appl Quotient Rule. The proos o the other three parts o the theorem are let as an eercise (see Eercise 9).

6 330_003.q 11/3/0 :3 PM Page 1 1 CHAPTER Dierentiation EXAMPLE Dierentiating Trigonometric Functions NOTE 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. a. b. Function tan sec Derivative 1 sec sec tan sec 1 sec 1 tan EXAMPLE 9 Dierent Forms o a Derivative Dierentiate both orms o First orm: Secon orm: To show that the two erivatives are equal, ou can write 1 cos sin sin cos cos sin 1 cos sin sin sin 1 cos cos sin 1 cos sin csc cot 1 cos sin csc cot csc 1 sin 1 sin cos sin csc csc cot. csc cot. 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. Eample 1 Eample 3 Eample Eample 5 Eample 9 Ater Dierentiating sin cos cos sin sin 1 cos cos sin Ater Simpliing 15 sin cos sin

7 330_003.q 11/3/0 :3 PM Page 15 SECTION.3 Prouct an Quotient Rules an Higher-Orer Derivatives 15 NOTE: The secon erivative o is the erivative o the irst erivative o. 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 given b 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 ollows. First erivative: Secon erivative: Thir erivative: Fourth erivative: nth erivative:,,,, n,,,,, n,,, 3 3,, n n,,, 3 3,, n n, D D D 3 D D n EXAMPLE 10 Fining the Acceleration Due to Gravit NASA THE MOON The moon s mass is kilograms, an Earth s mass is kilograms. The moon s raius is 1737 kilometers, an Earth s raius is 37 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 1971, 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 given b where st is the height in meters an t is the time in secons. What is the ratio o Earth s gravitational orce to the moon s? st 0.1t To in the acceleration, ierentiate the position unction twice. st 0.1t st 1.t st 1. Position unction Velocit unction Acceleration unction So, the acceleration ue to gravit on the moon is 1. 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

8 330_003.q 11/3/0 :3 PM Page 1 1 CHAPTER Dierentiation Eercises or Section.3 See or worke-out solutions to o-numbere eercises. In Eercises 1, use the Prouct Rule to ierentiate the unction. 1. g ht tt 3. gs s s 5. 3 cos. g sin hs s h 1 In Eercises 7 1, use the Quotient Rule to ierentiate the unction g sin 1. In Eercises 13 1, in an c In Eercises 19, complete the table without using the Quotient Rule Function Function cos sin Rewrite In Eercises 5 3, in the erivative o the algebraic unction gt t t 7 s 9. h hs 3 1 s 1 t cos t t 3 Dierentiate Value o c c 0 c 1 c 1 c c c Simpli c c is a constant c, 3. c c is a constant c, In Eercises 39 5, in the erivative o the trigonometric unction. 39. t t sin t 0. 1 cos 1. t cos t. t 3. tan. cot 5. gt t. hs 1 sec t 10 csc s s 31 sin 7.. sec cos 9. csc sin 50. sin cos 51. tan 5. sin cos 53. sin cos 5. h 5 sec In Eercises 55 5, use a computer algebra sstem to ierentiate the unction g 5. 1 sin In Eercises 59, evaluate the erivative o the unction at the given point. Use a graphing utilit to veri our result g Function 1 csc 1 csc tan cot ht sec t t Point 1, 1, 1. sin sin cos, 1, 3 g 1 1 sin sin 1 cos tan

9 330_003.q 11/3/0 :3 PM Page 17 SECTION.3 Prouct an Quotient Rules an Higher-Orer Derivatives 17 In Eercises 3, (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 , 1, 1 5.,. 1, 1, 7. tan,. sec,, 1 3, 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 71 an 7 are calle serpentines () = (, 5) In Eercises 73 7, etermine the point(s) at which the graph o the unction has a horizontal tangent line () = + (, 1) , 1, 3 ( 3, 3 ) Tangent Lines Fin equations o the tangent lines to the graph o 1 that are parallel to the line. 1 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 1, 5. 1 Then graph the unction an the tangent lines. () = () = (, 5), In Eercises 79 an 0, veri that the relationship between an g g, an eplain In Eercises 1 an, use the graphs o an g. Let p g an q g. 1. (a) Fin p1.. (a) Fin p. (b) Fin q (b) Fin q7. 3. Area The length o a rectangle is given b t 1 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 1 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. 3, sin 3, 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) 10, (b) 15, an (c) 0. What o these rates o change impl about increasing orer size?. Bole s Law This law states that i the temperature o a gas remains constant, its pressure is inversel proportional to its volume. Use the erivative to show that the rate o change o the pressure is inversel proportional to the square o the volume. 7. Population Growth A population o 500 bacteria is introuce into a culture an grows in number accoring to the equation Pt g g g t 50 t 5 sin 1 where t is measure in hours. Fin the rate at which the population is growing when t g

10 330_003.q 11/3/0 :3 PM Page 1 1 CHAPTER Dierentiation. Gravitational Force Newton s Law o Universal Gravitation states that the orce F between two masses, an m, is F Gm 1 m where G is a constant an is the istance between the masses. Fin an equation that gives an instantaneous rate o change o F with respect to. (Assume m 1 an m represent moving points.) 9. Prove the ollowing ierentiation rules. (a) (c) sec sec tan cot csc 90. 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. 91. Moeling Data The table shows the numbers n (in thousans) o motor homes sol in the Unite States an the retail values v (in billions o ollars) o these motor homes or the ears 199 through 001. The ear is represente b t, with t corresponing to 199. (Source: Recreation Vehicle Inustr Association) (a) Use a graphing utilit to in cubic moels or the number o motor homes sol nt an the total retail value vt o the motor homes. (b) (b) Graph each moel oun in part (a). (c) Fin A vtnt, then graph A. What oes this unction represent? () Interpret At in the contet o these ata. 9. 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. csc csc cot Year, t n v m 1 In Eercises 93 9, in the secon erivative o the unction sin 9. sec In Eercises 99 10, in the given higher-orer erivative ,, 101., 10. 1, Writing About Concepts 103. Sketch the graph o a ierentiable unction such that 0, < 0 or < <, an > 0 or < <. 10. Sketch the graph o a ierentiable unction such that > 0 an < 0 or all real numbers. In Eercises , use the given inormation to in. g 3 h 1 an an In Eercises 109 an 110, the graphs o,, an are shown on the same set o coorinate aes. Which is which? Eplain our reasoning. To print an enlarge cop o the graph, go to the website g h 105. g h 10. h 107. g 10. gh h r r h θ In Eercises , the graph o is shown. Sketch the graphs o an. To print an enlarge cop o the graph, go to the website (a) Show that h rcsc 1. (b) Fin the rate at which h is changing with respect to when (Assume r 390 miles.) 30.

11 330_003.q 11/3/0 :3 PM Page 19 SECTION.3 Prouct an Quotient Rules an Higher-Orer Derivatives Acceleration The velocit o an object in meters per secon is vt 3 t, 0 t. Fin the velocit an acceleration o the object when t 3. What can be sai about the spee o the object when the velocit an acceleration have opposite signs? 11. Acceleration An automobile s velocit starting rom rest is vt where v is measure in eet per secon. Fin the acceleration at (a) 5 secons, (b) 10 secons, an (c) 0 secons 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. t st vt at 11. Particle Motion The igure shows the graphs o the position, velocit, an acceleration unctions o a particle. 1 1 π 3π 100t t (a) Cop the graphs o the unctions shown. Ienti each graph. Eplain our reasoning. To print an enlarge cop o the graph, go to the website (b) On our sketch, ienti when the particle spees up an when it slows own. Eplain our reasoning. Fining a Pattern In Eercises 119 an 10, evelop a general rule or n given n t 1 1 π π 3π π 11. Fining a Pattern Consier the unction gh. (a) Use the Prouct Rule to generate rules or ining,, an. (b) Use the results in part (a) to write a general rule or n. 1. Fining a Pattern Develop a general rule or n where is a ierentiable unction o. In Eercises 13 an 1, in the erivatives o the unction or n 1,, 3, an. Use the results to write a general rule or in terms o n. 13. n sin 1. Dierential Equations In Eercises 15 1, veri that the unction satisies the ierential equation Function 1, > sin 3 3 cos sin True or False? In Eercises 19 13, etermine whether the statement is true or alse. I it is alse, eplain wh or give an eample that shows it is alse. 19. I g, then g I 1 3, then I c an gc are zero an h g, then hc I is an nth-egree polnomial, then n The secon erivative represents the rate o change o the irst erivative. 13. I the velocit o an object is constant, then its acceleration is zero Fin a secon-egree polnomial a b c such that its graph has a tangent line with slope 10 at the point, 7 an an -intercept at 1, Consier the thir-egree polnomial a 3 b c, Determine conitions or a, b, c, an i the graph o has (a) no horizontal tangents, (b) eactl one horizontal tangent, an (c) eactl two horizontal tangents. Give an eample or each case Fin the erivative o. Does 0 eist? 13. 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 (b) g g g Dierential Equation a 0. cos n