The Chain Rule. y x 2 1 y sin x. and. Rate of change of first axle. with respect to second axle. dy du. du dx. Rate of change of first axle
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1 . The Chain Rule 9. The Chain Rule Fin the erivative of a composite function using the Chain Rule. Fin the erivative of a function using the General Power Rule. Simplif the erivative of a function using algebra. Fin the erivative of a trigonometric function using the Chain Rule. The Chain Rule This tet has et to iscuss one of the most powerful ifferentiation rules the Chain Rule. This rule eals with composite functions an as a surprising versatilit to the rules iscusse in the two previous sections. For eample, compare the functions shown below. Those on the left can be ifferentiate without the Chain Rule, an those on the right are best ifferentiate with the Chain Rule. Without the Chain Rule sin tan With the Chain Rule sin 6 5 tan Basicall, the Chain Rule states that if changes u times as fast as u, an u changes u times as fast as, then changes uu times as fast as. The Derivative of a Composite Function Gear Gear Ale Gear Ale Gear Ale Ale : revolutions per minute Ale : u revolutions per minute Ale : revolutions per minute Figure. A set of gears is constructe, as shown in Figure., such that the secon an thir gears are on the same ale. As the first ale revolves, it rives the secon ale, which in turn rives the thir ale. Let, u, an represent the numbers of revolutions per minute of the first, secon, an thir ales, respectivel. Fin u, u, an, an show that u u. Solution Because the circumference of the secon gear is three times that of the first, the first ale must make three revolutions to turn the secon ale once. Similarl, the secon ale must make two revolutions to turn the thir ale once, an ou can write u an Combining these two results, ou know that the first ale must make si revolutions to turn the thir ale once. So, ou can write u u 6 u. Rate of change of first ale with respect to secon ale Rate of change of first ale with respect to thir ale. Rate of change of secon ale with respect to thir ale In other wors, the rate of change of with respect to is the prouct of the rate of change of with respect to u an the rate of change of u with respect to. 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 from the ebook an/or echapter(s). Eitorial review has
2 0 Chapter Differentiation Eploration Using the Chain Rule Each of the following functions can be ifferentiate using rules that ou stuie in Sections. an.. For each function, fin the erivative using those rules. Then fin the erivative using the Chain Rule. Compare our results. Which metho is simpler? a. b. c. sin REMARK The alternative limit form of the erivative was given at the en of Section.. Eample illustrates a simple case of the Chain Rule. The general rule is state in the net theorem. THEOREM.0 The Chain Rule If f u is a ifferentiable function of u an u g is a ifferentiable function of, then f g is a ifferentiable function of an u u or, equivalentl, f g fgg. Proof Let h f g. Then, using the alternative form of the erivative, ou nee to show that, for c, hc fgcgc. An important consieration in this proof is the behavior of g as approaches c. A problem occurs when there are values of, other than c, such that g gc. Appeni A shows how to use the ifferentiabilit of f an g to overcome this problem. For now, assume that g gc for values of other than c. In the proofs of the Prouct Rule an the Quotient Rule, the same quantit was ae an subtracte to obtain the esire form. This proof uses a similar technique multipling an iviing b the same (nonzero) quantit. Note that because g is ifferentiable, it is also continuous, an it follows that g approaches gc as approaches c. hc lim c f g f gc c f g f gc g gc lim c c f g f gc lim c g gc f g f gc lim c g gc lim c fgcgc Alternative form of erivative g gc g gc c g gc c, g gc See LarsonCalculus.com for Bruce Ewars s vieo of this proof. When appling the Chain Rule, it is helpful to think of the composite function as having two parts an inner part an an outer part. Outer function f g f g f u Inner function The erivative of f u is the erivative of the outer function (at the inner function u) times the erivative of the inner function. fu u 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 from the ebook an/or echapter(s). Eitorial review has
3 . The Chain Rule Decomposition of a Composite Function a. b. c.. fg sin tan u g u u u u tan fu u sin u u u Using the Chain Rule REMARK You coul also solve the problem in Eample without using the Chain Rule b observing that an Verif that this is the same as the erivative in Eample. Which metho woul ou use to fin 50? Fin for. Solution For this function, ou can consier the insie function to be u an the outer function to be u. B the Chain Rule, ou obtain 6. u u The General Power Rule The function in Eample is an eample of one of the most common tpes of composite functions, u n. The rule for ifferentiating such functions is calle the General Power Rule, an it is a special case of the Chain Rule. THEOREM. The General Power Rule If u n, where u is a ifferentiable function of an n is a rational number, then u nun or, equivalentl, un nu n u. Proof Because u n u n, ou appl the Chain Rule to obtain u u u un u. B the (Simple) Power Rule in Section., ou have D u u n nu n, an it follows that u nun. See LarsonCalculus.com for Bruce Ewars s vieo of this proof. 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 from the ebook an/or echapter(s). Eitorial review has
4 Chapter Differentiation Appling the General Power Rule Fin the erivative of f. Solution Let u. Then f u an, b the General Power Rule, the erivative is n u n u f. Appl General Power Rule. Differentiate. f() = ( ) f () = The erivative of f is 0 at 0 an is unefine at ±. Figure.5 Fin all points on the graph of f Differentiating Functions Involving Raicals for which f 0 an those for which f oes not eist. Solution Begin b rewriting the function as f. Then, appling the General Power Rule (with u prouces f n u n. u Appl General Power Rule. Write in raical form. So, f 0 when 0, an f oes not eist when ±, as shown in Figure.5. Differentiating Quotients: Constant Numerators REMARK Tr ifferentiating the function in Eample 6 using the Quotient Rule. You shoul obtain the same result, but using the Quotient Rule is less efficient than using the General Power Rule. Differentiate the function gt Solution 7 t. Begin b rewriting the function as gt 7t. Then, appling the General Power Rule (with u t ) prouces n u n u gt 7t Constant Multiple Rule 8t 8 t. Appl General Power Rule. Simplif. Write with positive eponent. 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 from the ebook an/or echapter(s). Eitorial review has
5 Simplifing Derivatives. The Chain Rule The net three eamples emonstrate techniques for simplifing the raw erivatives of functions involving proucts, quotients, an composites. Simplifing b Factoring Out the Least Powers Fin the erivative of f. Solution f f Write original function. Rewrite. Prouct Rule General Power Rule Simplif. Factor. Simplif. Simplifing the Derivative of a Quotient TECHNOLOGY Smbolic ifferentiation utilities are capable of ifferentiating ver complicate functions. Often, however, the result is given in unsimplifie form. If ou have access to such a utilit, use it to fin the erivatives of the functions given in Eamples 7, 8, an 9. Then compare the results with those given in these eamples. f f Original function Rewrite. Quotient Rule Factor. Simplif. Simplifing the Derivative of a Power See LarsonCalculus.com for an interactive version of this tpe of eample. n u n u Original function General Power Rule Quotient Rule Multipl. Simplif. 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 from the ebook an/or echapter(s). Eitorial review has
6 Chapter Differentiation Trigonometric Functions an the Chain Rule The Chain Rule versions of the erivatives of the si trigonometric functions are shown below. sin u cos uu tan u sec uu sec u sec u tan uu cos u sin uu cot u csc uu csc u csc u cot uu The Chain Rule an Trigonometric Functions u cos u u a. sin cos cos cos u sin u u b. cos sin sin u sec u u c. tan sec sec sec Be sure ou unerstan the mathematical conventions regaring parentheses an trigonometric functions. For instance, in Eample 0(a), sin is written to mean sin. a. b. e. cos cos Parentheses an Trigonometric Functions cos cos sin 6 6 sin cos cos cos c. cos cos9 sin sin 9. cos cos cos sin cos sin cos sin sin cos To fin the erivative of a function of the form k fgh, ou nee to appl the Chain Rule twice, as shown in Eample. Repeate Application of the Chain Rule ft sin t sin t ft sin t sin t t sin t cos t t t sin t cos t sin t cos t Original function Rewrite. Appl Chain Rule once. Appl Chain Rule a secon time. Simplif. 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 from the ebook an/or echapter(s). Eitorial review has
7 . The Chain Rule 5 f() = sin + cos Tangent Line of a Trigonometric Function (, ) Fin an equation of the tangent line to the graph of f sin cos at the point,, as shown in Figure.6. Then etermine all values of in the interval 0, at which the graph of f has a horizontal tangent. Solution Begin b fining f. f sin cos f cos sin cos sin Write original function. Appl Chain Rule to cos. Simplif. To fin the equation of the tangent line at,, evaluate f. Figure.6 f cos sin Substitute. Slope of graph at, Now, using the point-slope form of the equation of a line, ou can write m. Point-slope form Substitute for, m, an. Equation of tangent line at, You can then etermine that f 0 when an So, f has horizontal 6,, 6,. tangents at 6,, 5 an 6,. 5 This section conclues with a summar of the ifferentiation rules stuie so far. To become skille at ifferentiation, ou shoul memorize each rule in wors, not smbols. As an ai to memorization, note that the cofunctions (cosine, cotangent, an cosecant) require a negative sign as part of their erivatives. SUMMARY OF DIFFERENTIATION RULES General Differentiation Rules Let f, g, an u be ifferentiable functions of. Derivatives of Algebraic Functions Derivatives of Trigonometric Functions Chain Rule Constant Multiple Rule: cf cf Prouct Rule: fg fg gf Constant Rule: c 0 sin cos cos sin Chain Rule: fu fu u Sum or Difference Rule: f ± g f ± g Quotient Rule: g f gf fg g (Simple) Power Rule: n n n, tan sec cot csc General Power Rule: un nu n u sec sec tan csc csc cot 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 from the ebook an/or echapter(s). Eitorial review has
8 6 Chapter Differentiation. Eercises See CalcChat.com for tutorial help an worke-out solutions to o-numbere eercises. Decomposition of a Composite Function 6, complete the table fg tan csc sin 5 Fining a Derivative of the function. In Eercises In Eercises 7, fin the erivative g 9 0. f t 9t. 6. f f 5 9. f t 0. t. f. f g 5 0. u g. f v v. v Fining a Derivative Using Technolog In Eercises 5 0, use a computer algebra sstem to fin the erivative of the function. Then use the utilit to graph the function an its erivative on the same set of coorinate aes. Describe the behavior of the function that correspons to an zeros of the graph of the erivative cos 0. tan fu. f t 5 t. g st 5t t t.. gt 5 t ht t t g. f 5. g g Slope of a Tangent Line In Eercises an, fin the slope of the tangent line to the sine function at the origin. Compare this value with the number of complete ccles in the interval [0, ]. What can ou conclue about the slope of the sine function sin a at the origin?. (a). (a) = sin Fining a Derivative erivative of the function. In Eercises 6, fin the. cos. sin 5. g 5 tan 6. h sec 7. sin 8. cos 9. h sin cos f cot 5. sin 5. sec 5. gt 5 cos t 55. f tan g cos f sin 58. ht cot t 59. f t sec t cos 6. sin 6. sin sin 6. sintan 6. cossintan Evaluating a Derivative In Eercises 65 7, fin an evaluate the erivative of the function at the given point. Use a graphing utilit to verif our result ,, 66. 5, 67., f 5, 68., f 6, t 69. f t 0, 70. t, = sin 7. 6 sec 0, 5 7. cos,,, = sin g sec tan gv cos v csc v = sin f 5,, 9, 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 from the ebook an/or echapter(s). Eitorial review has
9 . The Chain Rule 7 Fining an Equation of a Tangent Line In Eercises 7 80, (a) fin an equation of the tangent line to the graph of f at the given point, use a graphing utilit to graph the function an its tangent line at the point, an (c) use the erivative feature of the graphing utilit to confirm our results. 7. f 7,, 5 7. f 5,, 75.,, 76. f 9,, 77. f sin,, cos,, 79. f tan,, 80. tan,, Famous Curves In Eercises 8 an 8, fin an equation of the tangent line to the graph at the given point. Then use a graphing utilit to graph the function an its tangent line in the same viewing winow. 8. Top half of circle 8. Bullet-nose curve 8. Horizontal Tangent Line Determine the point(s) in the interval 0, at which the graph of f cos sin has a horizontal tangent. 8. Horizontal Tangent Line Determine the point(s) at which the graph of f 8 6 f() = 5 (, ) 6 6 has a horizontal tangent. Fining a Secon Derivative secon erivative of the function. In Eercises 85 90, fin the 85. f f f 88. f f sin 90. f sec Evaluating a Secon Derivative In Eercises 9 9, evaluate the secon erivative of the function at the given point. Use a computer algebra sstem to verif our result., h 9. f 9,, 9. f cos, 0, 9. gt tan t, f() = (, ) 0, 6, WRITING ABOUT CONCEPTS Ientifing Graphs In Eercises 95 98, the graphs of a function f an its erivative are shown. Label the graphs as f or an write a short paragraph stating the criteria ou use in making our selection. To print an enlarge cop of the graph, go to MathGraphs.com Describing a Relationship In Eercises 99 an 00, the relationship between f an g is given. Eplain the relationship between an g. 99. g f 00. g f 0. Think About It The table shows some values of the erivative of an unknown function f. Complete the table b fining the erivative of each transformation of f, if possible. (a) h f (c) () s f 0. Using Relationships Given that g5, g5 6, h5, an h5, fin f5 for each of the following, if possible. If it is not possible, state what aitional information is require. (a) f g f r f f gh f f gh (c) f g () f g h f 0 f g h r s 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 from the ebook an/or echapter(s). Eitorial review has
10 8 Chapter Differentiation Fining Derivatives In Eercises 0 an 0, the graphs of f an g are shown. Let h fg an s g f. Fin each erivative, if it eists. If the erivative oes not eist, eplain wh. 09. Moeling Data The normal ail maimum temperatures T (in egrees Fahrenheit) for Chicago, Illinois, are shown in the table. (Source: National Oceanic an Atmospheric Aministration) 0. (a) Fin h. 0. (a) Fin h. Fin s f 6 g Fin s f g Month Jan Feb Mar Apr Temperature Month Ma Jun Jul Aug Temperature Month Sep Oct Nov Dec Temperature Doppler Effect The frequenc F of a fire truck siren hear b a stationar observer is F,00 ± v where ±v represents the velocit of the accelerating fire truck in meters per secon (see figure). Fin the rate of change of F with respect to v when (a) the fire truck is approaching at a velocit of 0 meters per secon (use v). the fire truck is moving awa at a velocit of 0 meters per secon (use v).,00,00 F = F = + v v 06. Harmonic Motion The isplacement from equilibrium of an object in harmonic motion on the en of a spring is cos t sin t where is measure in feet an t is the time in secons. Determine the position an velocit of the object when t Penulum A 5-centimeter penulum moves accoring to the equation where is the angular isplacement from the vertical in raians an t is the time in secons. Determine the maimum angular isplacement an the rate of change of when t secons. 0. cos 8t, 08. Wave Motion A buo oscillates in simple harmonic motion A cos t as waves move past it. The buo moves a total of.5 feet (verticall) from its low point to its high point. It returns to its high point ever 0 secons. (a) Write an equation escribing the motion of the buo if it is at its high point at t 0. Determine the velocit of the buo as a function of t. (a) Use a graphing utilit to plot the ata an fin a moel for the ata of the form Tt a b sinct where T is the temperature an t is the time in months, with t corresponing to Januar. Use a graphing utilit to graph the moel. How well oes the moel fit the ata? (c) Fin T an use a graphing utilit to graph the erivative. () Base on the graph of the erivative, uring what times oes the temperature change most rapil? Most slowl? Do our answers agree with our observations of the temperature changes? Eplain. 0. HOW DO YOU SEE IT? The cost C (in ollars) of proucing units of a prouct is C For one week, management etermine that the number of units prouce at the en of t hours can be moele b.6t 9t 0.5t. The graph shows the cost C in terms of the time t. Cost (in ollars) 5,000 0,000 5,000 0,000 5,000 C Cost of Proucing a Prouct 5 Time (in hours) (a) Using the graph, which is greater, the rate of change of the cost after hour or the rate of change of the cost after hours? Eplain wh the cost function is not increasing at a constant rate uring the eight-hour shift. 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 from the ebook an/or echapter(s). Eitorial review has
11 . The Chain Rule 9. Biolog The number N of bacteria in a culture after t as is moele b N 00 Fin the rate of change of N with respect to t when (a) t 0, t, (c) t, () t, an (e) t. (f) What can ou conclue?. Depreciation The value V of a machine t ears after it is purchase is inversel proportional to the square root of t. The initial value of the machine is $0,000. (a) Write V as a function of t. Fin the rate of epreciation when t. (c) Fin the rate of epreciation when t.. Fining a Pattern Consier the function f sin, where is a constant. (a) Fin the first-, secon-, thir-, an fourth-orer erivatives of the function. Verif that the function an its secon erivative satisf the equation f f 0. (c) Use the results of part (a) to write general rules for the even- an o-orer erivatives f k an f k. [Hint: k is positive if k is even an negative if k is o.]. Conjecture Let f be a ifferentiable function of perio p. (a) Is the function perioic? Verif our answer. Consier the function g f. Is the function g perioic? Verif our answer. 5. Think About It Let r f g an s g f, where f an g are shown in the figure. Fin (a) r an s. 6. Using Trigonometric Functions (a) Fin the erivative of the function in two was. For f sec an g tan, show that f g. t f (, ) g f (6, 6) (6, 5) g sin cos 7. Even an O Functions (a) Show that the erivative of an o function is even. That is, if f f, then f f. Show that the erivative of an even function is o. That is, if f f, then f f. 8. Proof Let u be a ifferentiable function of Use the fact that to prove that u u. u u u u, Using Absolute Value In Eercises 9, use the result of Eercise 8 to fin the erivative of the function. g 5 h cos Linear an Quaratic Approimations The linear an quaratic approimations of a function f at a are P fa a fa an P fa a fa a fa. In Eercises an, (a) fin the specifie linear an quaratic approimations of f, use a graphing utilit to graph f an the approimations, (c) etermine whether P or P is the better approimation, an () state how the accurac changes as ou move farther from a.. f tan ; a. f sec ; True or False? In Eercises 5 8, etermine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 5. If, then 6. If f sin, then f sin cos. 7. If is a ifferentiable function of u, an u is a ifferentiable function of, then is a ifferentiable function of. 8. If is a ifferentiable function of u, u is a ifferentiable function of v, an v is a ifferentiable function of, then u v u v. u 0. f 9 f sin. PUTNAM EXAM CHALLENGE a 6 9. Let f a sin a sin... a n sin n, where a, a,..., a n are real numbers an where n is a positive integer. Given that f sin for all real, prove that a a... na n. 0. Let k be a fie positive integer. The nth erivative P of has the form n where P n is a k k n polnomial. Fin P n. These problems were compose b the Committee on the Putnam Prize Competition. The Mathematical Association of America. All rights reserve. Tischenko Irina/Shutterstock.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 from the ebook an/or echapter(s). Eitorial review has
12 Answers to O-Numbere Eercises A Section. (page 6) f g u g 5 8 f u csc 9. u 5 8 u 7 u csc 08 9 u u u. 5 t t v. v sin cos 5 5 The zero of correspons to the point on the graph of the function where the tangent line is horizontal. has no zeros. The zeros of correspon to the points on the graph of the function where the tangent lines are horizontal.. (a) ; The slope of sin a at the origin is a.. sin 5. 5 sec cos 5. cos sin 5. 8 sec tan tan 5 sec sin cos sin 59. cos t 6. cos 6. sec costan f 5, 8, 5 5 cos 6 sint 69. ft 5 t, (a) (a) (a) (a) ,, 6,,, cos sin 6 9. h 8 6, 9. f cos sin, f f 6 6 (, 0) 0 8 The zeros of correspon The zeros of correspon to the points where the graph to the points where the graph of f has horizontal tangents. of f has horizontal tangents. 99. The rate of change of g is three times as fast as the rate of change of f. 0. (a) g f h f) (c) r f () s f f (, ) (, 5) 9 sec tan, 0 f (, ) 0 g h 8 8 r f f (, f ( s 0. (a) s5 oes not eist because g is not ifferentiable at (a) ra,.5 rasec 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 from the ebook an/or echapter(s). Eitorial review has
13 A Answers to O-Numbere Eercises 09. (a) Tt sin0.8t (c) 0 0 The moel is a goo fit. Tt.5 cos0.8t () The temperature changes most rapil aroun spring (March Ma) an fall (Oct. Nov.) The temperature changes most slowl aroun winter (Dec. Feb.) an summer (Jun. Aug.) Yes. Eplanations will var.. (a) 0 bacteria per a 77.8 bacteria per a (c). bacteria per a () 0.8 bacteria per a (e). bacteria per a (f) The rate of change of the population is ecreasing as time passes.. (a) f cos (c) 0 f sin f cos f sin f f sin sin 0 f k k k sin f k kk cos 5. (a) r 0 s (a) an Proofs g 5 5, h sin 0 cos,. (a) P P 5 (c) f P P P 0 p () The accurac worsens as ou move awa from. 5. False. If, then 7. True 9. Putnam Problem A, 967. 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 from the ebook an/or echapter(s). Eitorial review has
With the Chain Rule. y x 2 1. and. with respect to second axle. dy du du dx. Rate of change of first axle. with respect to third axle
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