2.3 Product and Quotient Rules and Higher-Order Derivatives

118 Chapter 2 Differentiation

Find the derivative of a function using the Product Rule. Find the derivative of a function using the Quotient Rule. Find the derivative of a trigonometric function. Find a higher-order derivative of a function.

The Product Rule In Section 2.2, you learned that the derivative of the sum of two functions is simply the sum of their derivatives. The rules for the derivatives of the product and quotient of two functions are not as simple.

THEOREM 2.7 The Product Rule REMARK A version of the The product of two differentiable functions and gf is itself differentiable. Product Rule that some people Moreover, the derivative of fg is the first function times the derivative of the prefer is second, plus the second function times the derivative of the first. d d ͓ f͑x͒g͑x͔͒ f͑x͒g͑x͒ f͑x͒g͑x͒. ͓ f͑x͒g͑x͔͒ f͑x͒g͑x͒ g͑x͒f͑x͒ dx dx The advantage of this form is that it generalizes easily to products of three or more Proof Some mathematical proofs, such as the proof of the Sum Rule, are straight- factors. forward. Others involve clever steps that may appear unmotivated to a reader. This proof involves such a step—subtracting and adding the same quantity—which is shown in color. d f͑x x͒g͑x x͒ f͑x͒g͑x͒ ͓ f͑x͒g͑x͔͒ lim dx x→0 x f͑x x͒g͑x x͒ f͑x x͒g͑x͒ f͑x x͒g͑x͒ f͑x͒g͑x͒ lim x→0 x g͑x x͒ g͑x͒ f͑x x͒ f͑x͒ lim ΄f͑x x͒ g͑x͒ ΅ x→0 x x g͑x x͒ g͑x͒ f͑x x͒ f͑x͒ lim ΄f͑x x͒ ΅ lim ΄g͑x͒ ΅ x→0 x x→0 x g͑x x͒ g͑x͒ f͑x x͒ f͑x͒ lim f͑x x͒ lim lim g͑x͒ lim x→0 x→0 x x→0 x→0 x f͑x͒g͑x͒ g͑x͒f͑x͒ Note that lim f ͑x x͒ f ͑x͒ because f is given to be differentiable and therefore x→0 is continuous. See LarsonCalculus.com for Bruce Edwards’s video of this proof.

The Product Rule can be extended to cover products involving more than two REMARK The proof of the factors. For example, if f, , and hg are differentiable functions of x, then Product Rule for products of d ͓ f͑x͒g͑x͒h͑x͔͒ f͑x͒g͑x͒h͑x͒ f͑x͒g͑x͒h͑x͒ f͑x͒g͑x͒h͑x͒. more than two factors is left as dx an exercise (see Exercise 137). So, the derivative of y x2 sin x cos x is dy 2x sin x cos x x2 cos x cos x x2 sin x͑sin x͒ dx 2x sin x cos x x2͑cos2 x sin2 x͒.

The derivative of a product of two functions is not (in general) given by the product THE PRODUCT RULE of the derivatives of the two functions. To see this, try comparing the product of the When Leibniz originally wrote a derivatives of formula for the Product Rule, he was motivated by the expression f͑x͒ 3x 2x2 ͑x dx͒͑y dy͒ xy and from which he subtracted dx dy g͑x͒ 5 4x (as being negligible) and obtained the differential form x dy y dx. with the derivative in Example 1. This derivation resulted in the traditional form of the Product Rule. (Source: The History of Using the Product Rule Mathematics by David M. Burton) Find the derivative of h͑x͒ ͑3x 2x2͒͑5 4x͒. Solution

Derivative of Derivative First second Second of first d d h͑x͒ ͑3x 2x2͒ ͓5 4x͔ ͑5 4x͒ ͓3x 2x2͔ Apply Product Rule. dx dx ͑3x 2x2͒͑4͒ ͑5 4x͒͑3 4x͒ ͑12x 8x2͒ ͑15 8x 16x2͒ 24x2 4x 15

In Example 1, you have the option of finding the derivative with or without the Product Rule. To find the derivative without the Product Rule, you can write ͓͑ 2͒͑ ͔͒ ͓ 3 2 ͔ Dx 3x 2x 5 4x Dx 8x 2x 15x 24x 2 4x 15. In the next example, you must use the Product Rule.

Using the Product Rule

Find the derivative of y 3x2 sin x. Solution d d d ͓3x2 sin x͔ 3x2 ͓sin x͔ sin x ͓3x2͔ Apply Product Rule. dx dx dx 3x2 cos x ͑sin x͒͑6x͒ 3x2 cos x 6x sin x 3x͑x cos x 2 sin x͒

REMARK In Example 3, Using the Product Rule notice that you use the Product Rule when both factors of the Find the derivative of y 2x cos x 2 sin x. product are variable, and you Solution use the Constant Multiple Rule when one of the factors is a Product Rule Constant Multiple Rule constant. dy d d d ͑2x͒΂ ͓cos x͔΃ ͑cos x͒΂ ͓2x͔΃ 2 ͓sin x͔ dx dx dx dx ͑2x͒͑sin x͒ ͑cos x͒͑2͒ 2͑cos x͒ 2x sin x

THEOREM 2.8 The Quotient Rule The quotient f͞g of two differentiable functions f and g is itself differentiable at all values of x for which g͑x͒ 0. Moreover, the derivative of f͞g is given by the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. d f͑x͒ g͑x͒f͑x͒ f͑x͒g͑x͒ ΄ ΅ , g͑x͒ 0 dx g͑x͒ ͓g͑x͔͒ 2 REMARK From the Quotient Rule, you can see that the derivative of a quotient is not Proof (in general) the quotient of As with the proof of Theorem 2.7, the key to this proof is subtracting and the derivatives. adding the same quantity. ͑ ͒ ͑ ͒ f x x f x d f͑x͒ g͑x x͒ g͑x͒ ΄ ΅ lim Definition of derivative dx g͑x͒ x→0 x g͑x͒f͑x x͒ f͑x͒g͑x x͒ lim x→0 xg͑x͒g͑x x͒ g͑x͒f͑x x͒ f͑x͒g͑x͒ f͑x͒g͑x͒ f͑x͒g͑x x͒ lim x→0 xg͑x͒g͑x x͒ g͑x͓͒ f͑x x͒ f͑x͔͒ f͑x͓͒g͑x x͒ g͑x͔͒ lim lim x→0 x x→0 x lim ͓g͑x͒g͑x x͔͒ x→0 f͑x x͒ f͑x͒ g͑x x͒ g͑x͒ g͑x͒΄ lim ΅ f͑x͒΄ lim ΅ x→0 x x→0 x TECHNOLOGY A graphing lim ͓g͑x͒g͑x x͔͒ utility can be used to compare x→0 the graph of a function with g͑x͒f͑x͒ f͑x͒g͑x͒ the graph of its derivative. ͓g͑x͔͒ 2 For instance, in Figure 2.22, Note that lim g͑x x͒ g͑x͒ because g is given to be differentiable and therefore the graph of the function in x→0 Example 4 appears to have is continuous. two points that have horizontal See LarsonCalculus.com for Bruce Edwards’s video of this proof. tangent lines. What are the values of y at these two points? Using the Quotient Rule −5x2 + 4x + 5 5x 2 y′ = Find the derivative of y . (x2 + 1)2 x2 1 6 Solution d d ͑x2 1͒ ͓5x 2͔ ͑5x 2͒ ͓x2 1͔ d 5x 2 dx dx −78 ΄ ΅ Apply Quotient Rule. dx x2 1 ͑x2 1͒2 ͑x2 1͒͑5͒ ͑5x 2͒͑2x͒ ͑ 2 ͒2 5x − 2 −4 x 1 y = x2 + 1 ͑5x2 5͒ ͑10x2 4x͒ ͑x2 1͒2 Graphical comparison of a function and its derivative 5x2 4x 5 Figure 2.22 ͑x2 1͒2

Note the use of parentheses in Example 4. A liberal use of parentheses is recommended for all types of differentiation problems. For instance, with the Quotient Rule, it is a good idea to enclose all factors and derivatives in parentheses, and to pay special attention to the subtraction required in the numerator. When differentiation rules were introduced in the preceding section, the need for rewriting before differentiating was emphasized. The next example illustrates this point with the Quotient Rule.

Rewriting Before Differentiating

3 ͑1͞x͒ Find an equation of the tangent line to the graph of f ͑x͒ at ͑1, 1͒. x 5 Solution Begin by rewriting the function. 3 ͑1͞x͒ f ͑x͒ Write original function. x 5 1 x΂3 ΃ x Multiply numerator and denominator by x. − 1 x͑x 5͒ 3 x f(x) = x + 5 3x 1 y Rewrite. x2 5x 5 4 Next, apply the Quotient Rule. 3 ͑x2 5x͒͑3͒ ͑3x 1͒͑2x 5͒ y = 1 f͑x͒ Quotient Rule (−1, 1) ͑x2 5x͒2 ͑3x2 15x͒ ͑6x2 13x 5͒ x −7−6 −5 −4 −3 −2 −1 123 ͑x2 5x͒2 2 −2 3x 2x 5 Simplify. −3 ͑x2 5x͒2 − 4 To find the slope at ͑1, 1͒, evaluate f͑1͒. −5 f͑1͒ 0 Slope of graph at ͑1, 1͒ The line y 1 is tangent to the graph of f͑x͒ at the point ͑1, 1͒. Then, using the point-slope form of the equation of a line, you can determine that the Figure 2.23 equation of the tangent line at ͑1, 1͒ is y 1. See Figure 2.23.

Not every quotient needs to be differentiated by the Quotient Rule. For instance, each quotient in the next example can be considered as the product of a constant times a function of x. In such cases, it is more convenient to use the Constant Multiple Rule.

Using the Constant Multiple Rule REMARK To see the benefit Original Function Rewrite Differentiate Simplify of using the Constant Multiple x2 3x 1 1 2x 3 Rule for some quotients, try a. y y ͑x2 3x͒ y ͑2x 3͒ y using the Quotient Rule to 6 6 6 6 differentiate the functions in 5x 4 5 5 5 b. y y x 4 y ͑4x3͒ y x3 Example 6—you should 8 8 8 2 obtain the same results, but 3͑3x 2x2͒ 3 3 6 with more work. c. y y ͑3 2x͒ y ͑2͒ y 7x 7 7 7 9 9 9 18 d. y y ͑x2͒ y ͑2x3͒ y 5x2 5 5 5x3

In Section 2.2, the Power Rule was proved only for the case in which the exponent n is a positive integer greater than 1. The next example extends the proof to include negative integer exponents.

Power Rule: Negative Integer Exponents

If n is a negative integer, then there exists a positive integer k such that n k. So, by the Quotient Rule, you can write d d 1 ͓xn͔ ΄ ΅ dx dx xk xk͑0͒ ͑1͒͑kxk1͒ Quotient Rule and Power Rule ͑xk͒2 0 kxk1 x2k kxk1 nxn 1. n k So, the Power Rule

d ͓xn͔ nxn 1 Power Rule dx is valid for any integer. In Exercise 71 in Section 2.5, you are asked to prove the case for which n is any rational number.

Derivatives of Trigonometric Functions Knowing the derivatives of the sine and cosine functions, you can use the Quotient Rule to find the derivatives of the four remaining trigonometric functions.

THEOREM 2.9 Derivatives of Trigonometric Functions d d ͓tan x͔ sec2 x ͓cot x͔ csc2 x dx dx d d ͓sec x͔ sec x tan x ͓csc x͔ csc x cot x dx dx

Proof Considering tan x ͑sin x͒͑͞cos x͒ and applying the Quotient Rule, you REMARK In the proof of obtain Theorem 2.9, note the use of d d sin x ͓tan x͔ ΄ ΅ the trigonometric identities dx dx cos x 2 x 2 x ͑ ͒͑ ͒ ͑ ͒͑ ͒ sin cos 1 cos x cos x sin x sin x 2 Apply Quotient Rule. and cos x cos2 x sin2 x 1 sec x . cos2 x cos x 1 These trigonometric identities cos2 x and others are listed in sec2 x. Appendix C and on the formula cards for this text. See LarsonCalculus.com for Bruce Edwards’s video of this proof. The proofs of the other three parts of the theorem are left as an exercise (see Exercise 87).

Differentiating Trigonometric Functions

See LarsonCalculus.com for an interactive version of this type of example. Function Derivative dy a. y x tan x 1 sec2 x dx b. y x sec x y x͑sec x tan x͒ ͑sec x͒͑1͒ ͑sec x͒͑1 x tan x͒

Different Forms of a Derivative REMARK Because of Differentiate both forms of trigonometric identities, the derivative of a trigonometric 1 cos x y csc x cot x. function can take many forms. sin x This presents a challenge when Solution you are trying to match your answers to those given in the 1 cos x First form: y back of the text. sin x ͑sin x͒͑sin x͒ ͑1 cos x͒͑cos x͒ y sin2 x sin2 x cos x cos2 x sin2 x 1 cos x sin2 x cos2 x 1 sin2 x Second form: y csc x cot x y csc x cot x csc2 x To show that the two derivatives are equal, you can write 1 cos x 1 cos x sin2 x sin2 x sin2 x 1 1 cos x ΂ ΃΂ ΃ sin2 x sin x sin x csc2 x csc x cot x.

The summary below shows that much of the work in obtaining a simplified form of a derivative occurs after differentiating. Note that two characteristics of a simplified form are the absence of negative exponents and the combining of like terms.

f͑x͒ After Differentiating f͑x͒ After Simplifying Example 1 ͑3x 2x2͒͑4͒ ͑5 4x͒͑3 4x͒ 24x2 4x 15 Example 3 ͑2x͒͑sin x͒ ͑cos x͒͑2͒ 2͑cos x͒ 2x sin x ͑x2 1͒͑5͒ ͑5x 2͒͑2x͒ 5x2 4x 5 Example 4 ͑x2 1͒2 ͑x2 1͒2 ͑x2 5x͒͑3͒ ͑3x 1͒͑2x 5͒ 3x2 2x 5 Example 5 ͑x2 5x͒2 ͑x2 5x͒2 ͑sin x͒͑sin x͒ ͑1 cos x͒͑cos x͒ 1 cos x Example 6 sin2 x sin2 x

Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 124 Chapter 2 Differentiation Higher-Order Derivatives Just as you can obtain a velocity function by differentiating a position function, you can obtain an acceleration function by differentiating a velocity function. Another way of looking at this is that you can obtain an acceleration function by differentiating a position function twice.

s͑t͒ Position function v͑t͒ s͑t͒ Velocity function a͑t͒ v͑t͒ s͑t͒ Acceleration function The function a͑t͒ is the second derivative of s͑t͒ and is denoted by s͑t͒. The second derivative is an example of a higher-order derivative. You can define derivatives of any positive integer order. For instance, the third derivative is the derivative of the second derivative. Higher-order derivatives are denoted as shown below. dy d First derivative: y, f͑x͒, , ͓ f͑x͔͒, D ͓y͔ dx dx x d 2y d 2 Second derivative: y, f͑x͒, , ͓ f͑x͔͒, D 2͓y͔ dx2 dx2 x REMARK The second d 3y d 3 derivative of a function is the Third derivative: y, f͑x͒, , ͓ f ͑x͔͒, D 3͓y͔ dx3 dx3 x derivative of the first derivative d 4 y d 4 of the function. Fourth derivative: y͑4͒, f ͑4͒͑x͒, , ͓ f͑x͔͒, D 4 ͓y͔ dx4 dx4 x Ӈ dny dn nth derivative: y͑n͒, f ͑n͒͑x͒, , ͓ f ͑x͔͒, D n͓y͔ dxn dxn x

Finding the Acceleration Due to Gravity

Because the moon has no atmosphere, a falling s object on the moon encounters no air resistance. In 1971, astronaut David Scott demonstrated that 3 a feather and a hammer fall at the same rate on s(t) = −0.81t2 + 2

the moon. The position function for each of these 2 falling objects is ͑ ͒ 2 s t 0.81t 2 1 where s͑t͒ is the height in meters and t is the time in seconds, as shown in the figure at the t 123 The moon’s mass is 7.349 1022 right. What is the ratio of Earth’s gravitational kilograms, and Earth’s mass is force to the moon’s? 24 5.976 10 kilograms.The moon’s Solution To find the acceleration, differentiate the position function twice. radius is 1737 kilometers, and Earth’s radius is 6378 kilometers. s͑t͒ 0.81t 2 2 Position function Because the gravitational force s͑t͒ 1.62t Velocity function on the surface of a planet is ͑ ͒ directly proportional to its mass s t 1.62 Acceleration function and inversely proportional to the So, the acceleration due to gravity on the moon is 1.62 meters per second per second. square of its radius, the ratio of Because the acceleration due to gravity on Earth is 9.8 meters per second per second, the gravitational force on Earth the ratio of Earth’s gravitational force to the moon’s is to the gravitational force on the moon is Earth’s gravitational force 9.8 ͑5.976 1024͒͞63782 Moon’s gravitational force 1.62 Ϸ 6.0. ͑7.349 1022͒͞17372 Ϸ 6.0. NASA

2.3 Exercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises.

Using the Product Rule In Exercises 1–6, use the Product Finding a Derivative In Exercises 25–38, find the derivative Rule to find the derivative of the function. of the algebraic function.

1.g͑x͒ ͑x2 3͒͑x2 4x͒ 2. y ͑3x 4͒͑x3 5͒ 4 3x x 2 x2 5x 6 25.f͑x͒ 26. f͑x͒ 2 2 3.h͑t͒ Ίt͑1 t2͒ 4. g͑s͒ Ίs͑s2 8͒ x 1 x 4 4 2 5.f ͑x͒ x3 cos x 6. g͑x͒ Ίx sin x 27.f ͑x͒ x΂1 ΃ 28. f͑x͒ x 4΂1 ΃ x 3 x 1 Using the Quotient Rule In Exercises 7–12, use the 3x 1 29.f ͑x͒ 30. f͑x͒ Ί3 x͑Ίx 3͒ Quotient Rule to find the derivative of the function. Ίx x 3t2 1 31.h͑s͒ ͑s3 2͒2 32. h͑x͒ ͑x2 3͒3 7.f ͑x͒ 8. g͑t͒ x2 1 2t 5 1 2 Ίx x2 x 9.h͑x͒ 10. f͑x͒ 33. f ͑x͒ x3 1 2Ίx 1 x 3 sin x cos t 2 1 11.g͑x͒ 12. f ͑t͒ 34. g͑x͒ x2΂ ΃ x2 t3 x x 1 35. f͑x͒ ͑2x3 5x͒͑x 3͒͑x 2͒ Finding and Evaluating a Derivative In Exercises 13–18, ͑ ͒ ͑ 3 ͒͑ 2 ͒͑ 2 ͒ find fͧxͨ and fͧcͨ. 36. f x x x x 2 x x 1 x2 c2 Function Value of c 37. f͑x͒ , c is a constant x2 c2 13. f͑x͒ ͑x3 4x͒͑3x 2 2x 5͒ c 0 c2 x 2 38. f͑x͒ , c is a constant 14. y ͑x2 3x 2͒͑x3 1͒ c 2 c2 x 2 x2 4 15. f ͑x͒ c 1 x 3 Finding a Derivative of a Trigonometric Function In Exercises 39–54, find the derivative of the trigonometric x 4 16. f͑x͒ c 3 function. x 4 ͑ ͒ 2 ͑͒ ͑ ͒ 39.f t t sin t 40. f 1 cos 17. f͑x͒ x cos x c 4 ͑ ͒ cos t ͑ ͒ sin x 41.f t 42. f x 3 sin x t x 18. f͑x͒ c x 6 43.f͑x͒ x tan x 44. y x cot x 1 45.g͑t͒ Ί4 t 6 csc t 46. h͑x͒ 12 sec x Using the Constant Multiple Rule In Exercises 19–24, x complete the table to find the derivative of the function without 3͑1 sin x͒ sec x using the Quotient Rule. 47.y 48. y 2 cos x x Function Rewrite Differentiate Simplify 49.y csc x sin x 50. y x sin x cos x x2 3x 19. y 51.f͑x͒ x 2 tan x 52. f͑x͒ sin x cos x 7 53.y 2x sin x x2 cos x 54. h͑͒ 5 sec tan 5x2 3 20. y 4 Finding a Derivative Using Technology In Exercises 6 55–58, use a computer algebra system to find the derivative of 21. y 7x2 the function. 10 22. y x 1 3 55. g͑x͒ ΂ ΃͑2x 5͒ 3x x 2 3͞2 4x x2 x 3 23. y 56. f͑x͒ ΂ ΃͑x2 x 1͒ x x2 1 2x 24. y 1͞3 57. g͑͒ x 1 sin sin 58. f͑͒ 1 cos

Evaluating a Derivative In Exercises 59–62, evaluate the 77. Tangent Lines Find equations of the tangent lines to the derivative of the function at the given point. Use a graphing graph of f ͑x͒ ͑x 1͒͑͞x 1͒ that are parallel to the line utility to verify your result. 2y x 6. Then graph the function and the tangent lines. Function Point 78. Tangent Lines Find equations of the tangent lines to the graph of f ͑x͒ x͑͞x 1͒ that pass through the point ͑1, 5͒. 1 csc x 59. y ΂ , 3΃ Then graph the function and the tangent lines. 1 csc x 6 60. f͑x͒ tan x cot x ͑1, 1͒ Exploring a Relationship In Exercises 79 and 80, verify .gͧxͨ, and explain the relationship between f and g ؍ that fͧxͨ sec t 1 61. h͑t͒ ΂, ΃ t 3x 5x 4 79. f ͑x͒ , g͑x͒ x 2 x 2 62. f͑x͒ sin x͑sin x cos x͒ ΂ , 1΃ 4 sin x 3x sin x 2x 80. f ͑x͒ , g͑x͒ x x Finding an Equation of a Tangent Line In Exercises 63–68, (a) find an equation of the tangent line to the graph of Evaluating Derivatives In Exercises 81 and 82, use the .fͧxͨ/gͧxͨ ؍ fͧxͨgͧxͨ and qͧxͨ ؍ f at the given point, (b) use a graphing utility to graph the graphs of f and g. Let pͧxͨ function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. 81. (a) Find p͑1͒. 82. (a) Find p͑4͒. (b) Find q͑4͒. (b) Find q͑7͒. 63. f͑x͒ ͑x3 4x 1͒͑x 2͒, ͑1, 4͒ y y 64. f͑x͒ ͑x 2͒͑x2 4͒, ͑1, 5͒ 10 10 x x 3 65.f͑x͒ , ͑5, 5͒ 66. f͑x͒ , ͑4, 7͒ 8 f 8 f x 4 x 3 6 67.f͑x͒ tan x, ΂ , 1΃ 68. f͑x͒ sec x, ΂ , 2΃ 4 g 4 3 g 2 2 Famous Curves In Exercises 69–72, find an equation of the x x −2 2 46810 −2 2 46810 tangent line to the graph at the given point. (The graphs in Exercises 69 and 70 are called Witches of Agnesi. The graphs in 83. Area The length of a rectangle is given by 6t 5 and its Exercises 71 and 72 are called serpentines.) height is Ίt, where t is time in seconds and the dimensions are 69.y 70. y in centimeters. Find the rate of change of the area with respect 27 6 6 f(x) = to time. 8 x2 + 9 f(x) = 84. Volume The radius of a right circular cylinder is given by 4 x2 + 4 4 Ίt 2 and its height is 1Ίt, where t is time in seconds and −3, 3 2 ( 2 ( the dimensions are in inches. Find the rate of change of the (2, 1) volume with respect to time. x x −4 −2 24 −4 −2 24 85. Inventory Replenishment The ordering and transporta- −2 −2 tion cost C for the components used in manufacturing a product is 71.y 72. y 16x 200 x 8 f(x) = 4 C 100΂ ΃, x 1 x2 + 16 3 x2 x 30 2 2, 4 4 ( 5 ( 1 where C is measured in thousands of dollars and x is the order x x 1342 size in hundreds. Find the rate of change of C with respect to 48 − − 8 x when (a) x 10, (b) x 15, and (c) x 20. What do these 2, x ( 5 ( f(x) = 4 rates of change imply about increasing order size? 2 − x + 6 8 86. Population Growth A population of 500 bacteria is introduced into a culture and grows in number according to the Horizontal Tangent Line In Exercises 73–76, determine equation the point(s) at which the graph of the function has a horizontal tangent line. 4t P͑t͒ 500΂1 ΃ 50 t2 2x 1 x2 73.f ͑x͒ 74. f ͑x͒ x2 x2 1 where t is measured in hours. Find the rate at which the x2 x 4 population is growing when t 2. 75.f ͑x͒ 76. f ͑x͒ x 1 x2 7

87. Proof Prove the following differentiation rules. Finding a Higher-Order Derivative In Exercises 99–102, d find the given higher-order derivative. (a) ͓sec x͔ sec x tan x dx 99. f͑x͒ x 2, f͑x͒ d ͓ ͔ 2 (b) csc x csc x cot x 100. f͑x͒ 2 , f͑x͒ dx x d ͑4͒ (c) ͓cot x͔ csc2 x 101. f͑x͒ 2Ίx, f ͑x͒ dx 102. f ͑4͒͑x͒ 2x 1, f ͑6͒͑x͒ 88. Rate of Change Determine whether there exist any values of x in the interval ͓0, 2͒ such that the rate of change Using Relationships In Exercises 103–106, use the given of f ͑x͒ sec x and the rate of change of g͑x͒ csc x are information to find fͧ2ͨ. equal. ؊2؍ and gͧ2ͨ 3 ؍ gͧ2ͨ 89. Modeling Data The table shows the health care 4 ؍ ؊1 and hͧ2ͨ؍ expenditures h (in billions of dollars) in the United States and hͧ2ͨ the population p (in millions) of the United States for the years 103. f͑x͒ 2g͑x͒ h͑x͒ 2004 through 2009. The year is represented by t, with t 4 corresponding to 2004. (Source: U.S. Centers for Medicare 104. f͑x͒ 4 h͑x͒ & Medicaid Services and U.S. Census Bureau) g͑x͒ 105. f͑x͒ h͑x͒ Year, t 456789 106. f͑x͒ g͑x͒h͑x͒ h 1773 1890 2017 2135 2234 2330 WRITING ABOUT CONCEPTS p 293 296 299 302 305 307 107. Sketching a Graph Sketch the graph of a differentiable function f such that f͑2͒ 0, f < 0 for (a) Use a graphing utility to find linear models for the health < x < 2, and f > 0 for 2 < x < . Explain how h͑t͒ p͑t͒ care expenditures and the population . you found your answer. (b) Use a graphing utility to graph each model found in part 108. Sketching a Graph Sketch the graph of a differen- (a). tiable function f such that f > 0 and f < 0 for all real (c) Find A h͑t͒͞p͑t͒, then graph A using a graphing utility. numbers x. Explain how you found your answer. What does this function represent? Identifying Graphs In Exercises 109 and 110, the graphs (d) Find and interpret A͑t͒ in the context of these data. of f, f, and f are shown on the same set of coordinate axes. 90. Satellites When satellites observe Earth, they can scan Identify each graph. Explain your reasoning. To print an only part of Earth’s surface. Some satellites have sensors that enlarged copy of the graph, go to MathGraphs.com. can measure the angle shown in the figure. Let h represent the satellite’s distance from Earth’s surface, and let r represent 109.y 110. y Earth’s radius. 2

r x x θ −2 −1 2 −1 3 −1 rh −2

(a) Show that h r͑csc 1͒. Sketching Graphs In Exercises 111–114, the graph of f is (b) Find the rate at which h is changing with respect to when shown. Sketch the graphs of f and f. To print an enlarged 30. (Assume r 3960 miles.) copy of the graph, go to MathGraphs.com.

Finding a Second Derivative In Exercises 91–98, find the 111.y 112. y second derivative of the function. 8 4 3 2 5 3 2 f 91.f͑x͒ x 2x 3x x 92. f͑x͒ 4x 2x 5x 4 4 3͞2 2 3 93.f͑x͒ 4x 94. f͑x͒ x 3x 2 x 2 x x 3x x −8 4 95.f͑x͒ 96. f͑x͒ − − 442 f − x 1 x 4 −2 4 97.f͑x͒ x sin x 98. f͑x͒ sec x

113.y 114. y Finding a Pattern In Exercises 119 and 120, develop a ͧnͨͧ ͨ ͧ ͨ 4 f 4 general rule for f x given f x . 3 f 2 1 2 119.f ͑x͒ xn 120. f ͑x͒ 1 x x 1 π π 3 x 2 2 π π π π 121. Finding a Pattern Consider the function f͑x͒ g͑x͒h͑x͒. − 3 2 1 2 2 − (a) Use the Product Rule to generate rules for finding f͑x͒, −4 2 f͑x͒, and f ͑4͒͑x͒. ͑ ͒ 115. Acceleration The velocity of an object in meters per (b) Use the results of part (a) to write a general rule for f n ͑x͒. second is 122. Finding a Pattern Develop a general rule for ͓xf͑x͔͒͑n͒, where f is a differentiable function of x. v͑t͒ 36 t 2 for 0 t 6. Find the velocity and acceleration of the object Finding a Pattern In Exercises 123 and 124, find the and 4. Use the ,3 ,2 ,1 ؍ when t 3. What can be said about the speed of the object derivatives of the function f for n when the velocity and acceleration have opposite signs? results to write a general rule for fͧxͨ in terms of n. 116. Acceleration The velocity of an automobile starting from cos x 123.f͑x͒ xn sin x 124. f͑x͒ rest is xn

͑ ͒ 100t Differential Equations In Exercises 125–128, verify that v t 2t 15 the function satisfies the differential equation. where v is measured in feet per second. Find the acceleration Function Differential Equation at (a) 5 seconds, (b) 10 seconds, and (c) 20 seconds. 1 125. y , x > 0 x3 y 2x2 y 0 117. Stopping Distance A car is traveling at a rate of 66 feet x per second (45 miles per hour) when the brakes are applied. 3 2 The position function for the car is s͑t͒ 8.25t 2 66t, 126. y 2x 6x 10 y xy 2y 24x where s is measured in feet and t is measured in seconds. Use 127. y 2 sin x 3 y y 3 this function to complete the table, and find the average 128. y 3 cos x sin x y y 0 velocity during each time interval. True or False? In Exercises 129–134, determine whether the t 01234 statement is true or false. If it is false, explain why or give an example that shows it is false. s͑t͒ dy ͑ ͒ 129. If y f͑x͒g͑x͒, then f͑x͒g͑x͒. v t dx a͑t͒ d5y 130. If y ͑x 1͒͑x 2͒͑x 3͒͑x 4͒, then 0. dx5 131. If f͑c͒ and g͑c͒ are zero and h͑x͒ f͑x͒g͑x͒, then h͑c͒ 0. 118. HOW DO YOU SEE IT? The figure shows the 132. If f͑x͒ is an n th-degree polynomial, then f ͑n1͒͑x͒ 0. graphs of the position, velocity, and acceleration 133. The second derivative represents the rate of change of the functions of a particle. first derivative. y 134. If the velocity of an object is constant, then its acceleration is 16 zero. 12 8 135. Absolute Value Find the derivative of f ͑x͒ xԽxԽ. Does 4 ͑ ͒ t f 0 exist? (Hint: Rewrite the function as a piecewise −11 4567 function and then differentiate each part.) 136. Think About It Let f and g be functions whose first and second derivatives exist on an interval I. Which of the following formulas is (are) true? (a) Copy the graphs of the functions shown. Identify (a)fg f g ͑ fg fg͒ (b) fg fg ͑ fg͒ each graph. Explain your reasoning. To print an enlarged copy of the graph, go to MathGraphs.com. 137. Proof Use the Product Rule twice to prove that if f, g, and h are differentiable functions of x, then (b) On your sketch, identify when the particle speeds up and when it slows down. Explain your reasoning. d ͓ f͑x͒g͑x͒h͑x͔͒ f͑x͒g͑x͒h͑x͒ f͑x͒g͑x͒h͑x͒ f͑x͒g͑x͒h͑x͒. dx

Section 2.3 (page 125) 71. 25y 12x 16 0 73.͑1, 1͒ 75. ͑0, 0͒, ͑2, 4͒ 1. 2͑2x3 6x2 3x 6͒ 3. ͑1 5t2͒͑͞2Ίt ͒ 2͑ ͒ ͑ 2͒͑͞ 2 ͒2 77. Tangent lines: 2y x 7; 2y x 1 5. x 3 cos x x sin x 7. 1 x x 1 y ͑ 3͒͞ Ί ͑ 3 ͒2 ͑ ͒͞ 3 2y + x = 7 x + 1 9. 1 5x ͓2 x x 1 ͔ 11. x cos x 2 sin x x f(x) = − 6 x 1 13. f͑x͒ ͑x3 4x͒͑6x 2) ͑3x2 2x 5͒͑3x2 4͒ 4 3 2 (3, 2) 15x 8x 21x 16x 20 (−1, 0)

f͑0͒ 20 x − − − 2 6 4 2 224 x 6x 4 −2 15. f͑x͒ 17. f͑x͒ cos x x sin x ͑x 3͒2 −4 1 Ί2 −6 2y + x = −1 f͑1͒ f ΂ ΃ ͑4 ͒ 4 4 8 79. f͑x͒ 2 g͑x͒ 81. (a) p͑1͒ 1 (b) q͑4͒ 1͞3 Function Rewrite Differentiate Simplify 83. ͑18t 5͒͑͞2Ίt͒ cm2͞sec x2 3x 1 3 2 3 2x 3 85. (a) $38.13 thousand͞100 components 19. y y x2 x y x y 7 7 7 7 7 7 (b) $10.37 thousand͞100 components 6 6 12 12 ͞ 21. y y x2 y x3 y (c)$3.80 thousand 100 components 7x2 7 7 7x3 The cost decreases with increasing order size. 4x3͞2 2 23. y y 4x1͞2, y 2x1͞2 y , 87. Proof x Ίx 89. (a) h͑t͒ 112.4t 1332 x > 0 x > 0 p͑t͒ 2.9t 282 3 25. , x 1 27. ͑x2 6x 3͒͑͞x 3͒2 (b) 3000 400 ͑x 1͒2 h(t) p(t) 29. ͑3x 1͒͑͞2x3͞2͒ 31. 6s2͑s3 2͒ 33. ͑2x2 2x 3͓͒͞x2͑x 3͒2͔ 35. 4 3 2 10x 8x 21x 10x 30 2 10 2 10 4xc2 0 0 37. 39. t͑t cos t 2 sin t͒ ͑ 2 2͒2 112.4t 1332 x c (c) A 2.9t 282 2 2 2 41. ͑t sin t cos t͒͞t 43. 1 sec x tan x 10 1 3 45. 6 csc t cot t 47. sec x͑tan x sec x͒ 4t3͞4 2 49. cos x cot2 x 51. x͑x sec2 x 2 tan x͒ ͑ 2͒ 53. 4x cos x 2 x sin x 2 10 2x2 8x 1 1 sin cos 0 55. 57. ͑x 2͒2 ͑1 sin ͒2 A represents the average health care expenditures per 2 csc x cot x person (in thousands of dollars). 59. y , 4Ί3 ͑1 csc x͒2 27,834 61. h͑t͒ sec t͑t tan t 1͒͞t2, 1͞ 2 (d) A͑t͒ 8.41t2 1635.6t 79,524 63. (a) y 3x 1 65. (a) y 4x 25 A͑t͒ represents the rate of change of the average health 3 8 (b) (b) care expenditures per person for the given year t. (−5, 5) −13 −81

(1, −4) −6 −6 67. (a) 4x 2y 2 0 69. 2y x 4 0 (b) 4 π ( , 1 ( 4 −

−4

91. 12x2 12x 6 93.3͞Ίx 95. 2͑͞x 1͒3 125. y 1͞x2, y 2͞x3, 97. 2 cos x x sin x 99.2x 101.1͞Ίx 103. 0 x3y 2x2y x3͑2͞x3͒ 2x2͑1͞x2͒ 105. 10 2 2 0 107. Answers will vary. 109. y 127. y 2 cos x, y 2 sin x, f ′ Sample answer: 2 y y 2 sin x 2 sin x 3 3 f͑x͒ ͑x 2͒2 ͞ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ 1 f 129. False. dy dx f x g x g x f x 131. True y 133. True 135. f͑x͒ 2ԽxԽ; f͑0͒ does not exist. x 4 −2 −1 1 2 137. Proof

2 f ″ 1

x 1 2 34

111. y 113. y 4 f ′ 3 f ′ f ″ 2 1 1 x − − − x 3 2 1 12345 π π − 2 1 2 f ″ − −3 2 −4 −3 −5 −4

115. v͑3͒ 27 m͞sec a͑3͒ 6 m͞sec2 The speed of the object is decreasing. 117. t 01234 s͑t͒ 0 57.75 99 123.75 132 v͑t͒ 66 49.5 33 16.5 0

a͑t͒ 16.5 16.5 16.5 16.5 16.5 The average velocity on ͓0, 1͔ is 57.75, on ͓1, 2͔ is 41.25, on ͓2, 3͔ is 24.75, and on ͓3, 4͔ is 8.25. 119. f ͑n͒͑x͒ n͑n 1͒͑n 2͒ . . . ͑2͒͑1͒ n! 121. (a) f͑x͒ g͑x͒h͑x͒ 2g͑x͒h͑x͒ g͑x͒h͑x͒ f͑x͒ g͑x͒h͑x͒ 3g͑x͒h͑x͒ 3g͑x͒h͑x͒ g͑x͒h͑x͒ f ͑4͒͑x͒ g͑x͒h͑4͒͑x͒ 4g͑x͒h͑x͒ 6g͑x͒h͑x͒ 4g͑x͒h͑x͒ g͑4͒͑x͒h͑x͒ n! (b) f ͑n͒͑x͒ g͑x͒h͑n͒͑x͒ g͑x͒h͑n1͒͑x͒ 1!͑n 1͒! n! g͑x͒h͑n2͒͑x͒ . . . 2!͑n 2͒! n! g͑n1͒͑x͒h͑x͒ g͑n͒͑x͒h͑x͒ ͑n 1͒!1! 123. n 1: f͑x͒ x cos x sin x n 2: f͑x͒ x2 cos x 2x sin x n 3: f͑x͒ x3 cos x 3x2 sin x n 4: f͑x͒ x 4 cos x 4x3 sin x General rule: f͑x͒ x n cos x nx͑n1͒ sin x