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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1 0 CHAPTER Differentiation Section 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 one 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 Vieo Gear Gear Ale Gear Ale Gear Ale Ale : revolutions per minute Ale : u revolutions per minute Ale : revolutions per minute Figure Animation EXAMPLE The Derivative of a Composite Function 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 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 Rate of change of first ale with respect to secon ale u 6 u Rate of change of secon ale with respect to thir ale Rate of change of first 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 Tr It Eploration A

2 SECTION The Chain Rule EXPLORATION 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 Eample illustrates a simple case of the Chain Rule The general rule is state below 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 if 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 gc as c hc lim c f g f gc c f g f gc g gc lim c g gc c, f g f gc lim c g gc lim c fgcgc g gc c g gc When appling the Chain Rule, it is helpful to think of the composite function f g as having two parts an inner part an an outer part Outer function 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

3 CHAPTER Differentiation EXAMPLE Decomposition of a Composite Function fg u g fu a b c sin tan u u u u tan u sin u u u Tr It Eploration A EXAMPLE Using the Chain Rule STUDY TIP 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 B the Chain Rule, ou obtain 6 u u Tr It Eploration A Eploration B The eitable graph feature below allows ou to eit the graph of a function an its erivative Eitable Graph 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 Vieo Proof Because 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

4 SECTION The Chain Rule Vieo EXAMPLE 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() = ( ) f () = The erivative of f is 0 at 0 an is unefine at ± Figure 5 Eitable Graph f Tr It Appl General Power Rule Differentiate The eitable graph feature below allows ou to eit the graph of a function Eitable Graph EXAMPLE 5 Differentiating Functions Involving Raicals Fin all points on the graph of f 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 Tr It Eploration A Eploration A EXAMPLE 6 Differentiating Quotients with Constant Numerators NOTE 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 gt 7 t Solution Begin b rewriting the function as gt 7t Then, appling the General Power Rule prouces n u n gt 7t u Appl General Power Rule Constant Multiple Rule 8t 8 t Simplif Write with positive eponent Tr It Eploration A Eploration B

5 CHAPTER Differentiation Simplifing Derivatives The net three eamples illustrate some techniques for simplifing the raw erivatives of functions involving proucts, quotients, an composites EXAMPLE 7 Simplifing b Factoring Out the Least Powers f f Original function Rewrite Prouct Rule General Power Rule Simplif Factor Simplif Tr It EXAMPLE 8 Eploration A 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 on this page f f Original function Rewrite Quotient Rule Factor Simplif Tr It EXAMPLE 9 Eploration A Simplifing the Derivative of a Power n u n u Original function General Power Rule Quotient Rule Multipl Simplif Tr It Eploration A Open Eploration

6 SECTION The Chain Rule 5 Trigonometric Functions an the Chain Rule The Chain Rule versions of the erivatives of the si trigonometric functions are as shown sin u cos u u tan u sec u u sec u sec u tan u u cos u sin u u cot u csc u u csc u csc u cot u u Technolog EXAMPLE 0 Appling the Chain Rule to Trigonometric Functions u cos u u a sin b c cos tan cos cos cos sin sec Tr It Eploration A Be sure that ou unerstan the mathematical conventions regaring parentheses an trigonometric functions For instance, in Eample 0(a), sin is written to mean sin EXAMPLE Parentheses an Trigonometric Functions a b c e cos cos sin 6 6 sin cos cos cos9 sin sin 9 cos cos cos sin cos cos cos cos cos sin cos sin sin cos Tr It Eploration A To fin the erivative of a function of the form k fgh, ou nee to appl the Chain Rule twice, as shown in Eample EXAMPLE Repeate Application of the Chain Rule ft sin t sin t ft sin t Tr It sin t t sin t cos t t t sin t cos t sin t cos t Eploration A Original function Rewrite Appl Chain Rule once Appl Chain Rule a secon time Simplif

7 6 CHAPTER Differentiation EXAMPLE Tangent Line of a Trigonometric Function Figure 6 f() = sin + cos (, ) STUDY TIP To become skille at ifferentiation, ou shoul memorize each rule As an ai to memorization, note that the cofunctions (cosine, cotangent, an cosecant) require a negative sign as part of their erivatives 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 Write original function f cos sin Appl Chain Rule to cos cos sin Simplif To fin the equation of the tangent line at,, evaluate f 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 a 6,, 6, horizontal tangent at an 6,, 6, Tr It 5 Eploration A This section conclues with a summar of the ifferentiation rules stuie so far 5 Summar 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

8 SECTION The Chain Rule 7 Eercises for Section The smbol Click on Click on inicates an eercise in which ou are instructe to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarge cop of the graph In Eercises 6, complete the table 6 In Eercises 7, fin the erivative of the function g 9 0 f t 9t f t t g 5 9 g 5 6 f st t t 9 f t 0 5 t t gt t f f fg 6 5 tan 5 csc cos g 5 ht t t f v v v g u g fu In Eercises 8, 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 5 6 g 7 cos 8 tan In Eercises 9 an 0, 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? 9 (a) (b) 0 (a) (b) = sin = sin In Eercises 58, fin the erivative of the function cos sin g tan h sec 5 sin 6 cos 7 h sin cos 8 9 f cot 50 gv cos v sin csc v 5 sec 5 gt 5 cos t 5 f sin 5 ht cot t 55 f t sec t 56 5 cos 57 sin 58 sin sin In Eercises 59 66, evaluate the erivative of the function at the given point Use a graphing utilit to verif our result Function Point st t t 8 5,, 6, f f t f t t f 7 sec cos, 6 0,, 0, 6, g sec tan 5 = sin = sin

9 8 CHAPTER Differentiation In Eercises 67 7, (a) fin an equation of the tangent line to the graph of f at the given point, (b) 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 In Eercises 75 78, (a) use a graphing utilit to fin the erivative of the function at the given point, (b) fin an equation of the tangent line to the graph of the function at the given point, an (c) use the utilit to graph the function an its tangent line in the same viewing winow Function 67 f 68 f f 9 7 f sin cos f tan tan gt f, s t t t t, t t, t 9t, Famous Curves In Eercises 79 an 80, 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 79 Top half of circle 80 Bullet-nose curve 8 6 f() = 5 (, ) 6 6, 8, 0, 0, Point, 5,,,, 0 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 has a horizontal tangent,,, f() = (, ) In Eercises 8 86, fin the secon erivative of the function 8 f 8 85 f sin 86 f sec In Eercises 87 90, evaluate the secon erivative of the function at the given point Use a computer algebra sstem to verif our result h 9, f, 6, 9 0, 89 f cos, 0, 90 gt tan t, 6, Writing About Concepts f In Eercises 9 9, 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 use in making the selection To print an enlarge cop of the graph, select the MathGraph button f In Eercises 95 an 96, the relationship between f an g is given Eplain the relationship between an g 95 g f 96 g f 97 Given that g5, g5 6, h5, an h5, fin f5 (if possible) for each of the following If it is not possible, state what aitional information is require (a) f gh (b) f gh (c) f g () f g h f f

10 SECTION The Chain Rule 9 98 Think About It The table shows some values of the erivative of an unknown function f Complete the table b fining (if possible) the erivative of each transformation of f (a) g f (b) h f (c) r f () s f f g h r s In Eercises 99 an 00, the graphs of f an g are shown Let h f g an s g f Fin each erivative, if it eists If the erivative oes not eist, eplain wh 99 (a) Fin h 00 (a) Fin h (b) Fin s5 (b) Fin s Doppler Effect The frequenc F of a fire truck siren hear b a stationar observer is F,00 ± v f g 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) (b) the fire truck is moving awa at a velocit of 0 meters per secon (use v),00,00 F = F = + v v 0 8 f g 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 8 0 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 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 (b) Determine the velocit of the buo as a function of t 05 Circulator Sstem The spee S of bloo that is r centimeters from the center of an arter is S CR r where C is a constant, R is the raius of the arter, an S is measure in centimeters per secon Suppose a rug is aministere an the arter begins to ilate at a rate of Rt At a constant istance r, fin the rate at which S changes with respect to t for C , R 0, an Rt Moeling Data The normal ail maimum temperatures T (in egrees Fahrenheit) for Denver, Colorao, are shown in the table (Source: National Oceanic an Atmospheric Aministration) (a) Use a graphing utilit to plot the ata an fin a moel for the ata of the form Tt a b sint6 c where T is the temperature an t is the time in months, with t corresponing to Januar (b) Use a graphing utilit to graph the moel How well oes the moel fit the ata? (c) Fin 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 T 0 cos 8t, Month Jan Feb Mar Apr Ma Jun Temperature Month Jul Aug Sep Oct Nov Dec Temperature

11 0 CHAPTER Differentiation 07 Moeling Data The cost of proucing units of a prouct is C For one week management etermine the number of units prouce at the en of t hours uring an eight-hour shift The average values of for the week are shown in the table t (a) Use a graphing utilit to fit a cubic moel to the ata (b) Use the Chain Rule to fin Ct (c) Eplain wh the cost function is not increasing at a constant rate uring the 8-hour shift 08 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 (b) Verif that the function an its secon erivative satisf the equation f f 0 (c) Use the results in 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] 09 Conjecture Let f be a ifferentiable function of perio p (a) Is the function perioic? Verif our answer (b) Consier the function g f Is the function g perioic? Verif our answer 0 Think About It Let r f g an s g f where f an g are shown in the figure Fin (a) r an (b) s (, ) g (a) Fin the erivative of the function g sin cos in two was (b) For f sec an g tan, show that f g f f (6, 6) (6, 5) (a) Show that the erivative of an o function is even That is, if f f, then f f (b) Show that the erivative of an even function is o That is, if f f, then f f Let u be a ifferentiable function of Use the fact that to prove that In Eercises 7, use the result of Eercise to fin the erivative of the function u u u u u u, g f h cos f sin Linear an Quaratic Approimations The linear an quaratic approimations of a function f at a are P fa a f a an P f a a fa a f a) In Eercises 8 an 9, (a) fin the specifie linear an quaratic approimations of f, (b) 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 8 f tan 9 f sec a True or False? In Eercises 0, etermine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 0 If, then If f sin, then f sin cos 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 a 6 Putnam Eam Challenge 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 for all real prove that a a f sin, na n Let k be a fie positive integer The nth erivative of k has the form P n k n where P n is a polnomial Fin P n These problems were compose b the Committee on the Putnam Prize Competition The Mathematical Association of America All rights reserve

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

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 . 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