41. The ancient Babylonians (circa 1700 B.C.) approximated N by applying the formula. x n x n N x n. with respect to output

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1 48 Chapter 2 Differentiation: Basic Concepts 4. The ancient Babylonians (circa 700 B.C.) approximated N by applying the formula x n 2 x n N x n for n, 2, 3,... (a) Show that this formula can be derived from the formula for Newton s method in Problem 38, then use it to estimate,265. Repeat the formula until two consecutive approximations agree to four decimal places. Use your calculator to check your result. (b) The spy wakes up one morning in Babylonia finds that his calculator has been stolen. Create a spy story problem based on using the ancient formula to compute a square root. 5 The Chain Rule 42. Sometimes Newton s method fails no matter what initial value x 0 is chosen (unless we are lucky enough to choose the root itself). Let f(x) 3 x choose x 0 arbitrarily (x 0 0). (a) Show that x n 2x n for n 0,, 2,... so that the successive guesses generated by Newton s method are x 0, 2x 0, 4x 0,... (b) Use your graphing utility to graph f(x) use an appropriate utility to draw the tangent lines to the graph of y f(x) at the points that correspond to x 0, 2x 0, 4x 0,... Why do these numbers fail to estimate a root of f(x) 0? In many practical situations, you find that the rate at which one quantity is changing can be expressed as the product of other rates. For example, suppose a car is traveling at 50 mph at a particular time when gasoline is being consumed at the rate of 0. gal/mile. Then, to find out how much gasoline is being used each hour, you would multiply the rates: (0. gal/mile)(50 miles/hour) 5 gal/hour Or, suppose the total manufacturing cost at a certain factory is a function of the number of units produced, which in turn is a function of the number of hours the factory has been operating. If C, q, t denote the cost, units produced, time, respectively, then dc (dollars per unit) dq rate of change of cost with respect to output dq dt rate of change of output with respect to time (units per hour) The product of these two rates is the rate of change of cost with respect to time; that is,

2 Chapter 2 Section 5 The Chain Rule 49 dc dt dc dq dq dt (dollars per hour) This formula is a special case of an important result in calculus called the chain rule. The Chain Rule Suppose y is a differentiable function of u, u is a differentiable function of x. Then y is a composite function of x du du That is, the derivative of y with respect to x is the derivative of y with respect to u times the derivative of u with respect to x. Note Notice that one way to remember the chain rule is to pretend that the derivatives are quotients to cancel du. That is, du du du du Here are two examples illustrating the use of the chain rule. Find if y u 3 3u 2 u x 2 2. Since EXAMPLE 5. it follows that du 3u2 6u Notice that this derivative is expressed in terms of the variables x u. Since you are thinking of y as a function of x, you may want to express in terms of x alone. To do this, substitute x 2 2 for u in the expression for simplify the answer as follows. du 2x du du (3u2 6u)(2x)

3 50 Chapter 2 Differentiation: Basic Concepts [3(x2 2) 2 6(x 2 2)](2x) 6x(x 2 2)[(x 2 2) 2] 6x(x 2 2)(x 2 ) 6x 3 (x 2 2) For practice, check this answer by first substituting u x 2 2 into the original expression for y then differentiating with respect to x. In the next example, you will see how to use the chain rule to calculate a derivative for a particular value of the independent variable. EXAMPLE 5.2 Find when x if y u u 3x 2. u Since (u )() u() du (u ) 2 (u ) 2 by the quotient rule it follows that du 6x du du 6x (u ) 2(6x) (u ) 2 The goal is to evaluate this derivative when x. One way to do this is to replace u by its algebraic formula, as in Example 5., then evaluate the resulting expression when x. However, it is easier to substitute numbers than algebraic expressions, so it is preferable to compute the numerical value of u first then substitute. In particular, when x, the original formula u 3x 2 gives u 3() 2 2. Now substitute x u 2 in the formula for to conclude that when x, 6() (2 ) In many practical problems, a quantity is given as a function of one variable, which, in turn, can be written as a function of a second variable, the goal is to find the rate of change of the original quantity with respect to the second variable. Such problems can be solved by means of the chain rule. Here is an example.

4 Chapter 2 Section 5 The Chain Rule 5 The cost of producing x units of a particular commodity is C(x) x 2 4x 53 3 dollars, the production level t hours into a particular production run is x(t) 0.2t t units. At what rate is cost changing with respect to time after 4 hours? so according to the chain rule, When t 4, the level of production is EXAMPLE 5.3 dc 2 3 x 4 dc dt 0.4t 0.03 dt dc dt 2 3 x 4 (0.4t 0.03) x(4) 0.2(4) (4) 3.32 units dc dt 2 3 (3.32 4) [0.4(4) 0.03] Thus, after 4 hours, cost is increasing at the rate of approximately $0.3 per hour. COMPOSITE FUNCTIONS Recall from Chapter, Section, that the composite function g[h(x)] is the function formed from functions g(u) h(x) by substituting h(x) for u in the formula for g(u). The chain rule is actually a rule for differentiating composite functions can be rewritten using functional notation as follows. The Chain Rule in Functional Notation If g(u) h(x) are differentiable functions, d g[h(x)] g[h(x)]h(x) To see that this is nothing more than a restatement of the previous version of the chain rule, suppose that y g[h(x)]. Then, y g(u) where u h(x)

5 52 Chapter 2 Differentiation: Basic Concepts, by the chain rule, The use of this form of the chain rule is illustrated in the next example. EXAMPLE 5.4 Differentiate the function f(x) x 2 3x 2. The form of the function is where the box du g(u)h(x) g[h(x)]h(x) du contains the expression x 2 3x 2. Then according to the chain rule, the derivative of the composite function f(x) is f(x) 2 ( )/2 ( ) f(x) ( ) /2 ( ) (x 2 3x 2) 2x 3 2 ( )/2 (2x 3) 2 (x2 3x 2) /2 (2x 3) 2x 3 2x 2 3x 2 THE GENERAL POWER RULE In Section 2, you learned the rule d (xn ) nx n for differentiating power functions. By combining this rule with the chain rule, you obtain the following rule for differentiating functions of the general form [h(x)] n. The General Power Rule For any real number n differentiable function h, d [h(x)]n n[h(x)] n d [h(x)]

6 Chapter 2 Section 5 The Chain Rule 53 To derive the general power rule, think of [h(x)] n as the composite function [h(x)] n g[h(x)] where g(u) u n Then, g(u) nun h(x) d [h(x)], by the chain rule, d [h(x)]n d g[h(x)] g[h(x)]h(x) d n[h(x)]n [h(x)] The use of the general power rule is illustrated in the following examples. EXAMPLE 5.5 Differentiate the function f(x) (2x 4 x) 3. One way to do this problem is to exp the function rewrite it as f(x) 8x 2 2x 9 6x 6 x 3 then differentiate this polynomial term by term to get f(x) 96x 08x 8 36x 5 3x 2 But see how much easier it is to use the general power rule. According to this rule, f(x) 3(2x 4 x) 2 d (2x4 x) 3(2x 4 x) 2 (8x 3 ) Not only is this method easier, but the answer even comes out in factored form! In the next example, the solution to Example 5.4 is written more compactly with the aid of the general power rule. EXAMPLE 5.6 Differentiate the function f(x) x 2 3x 2. Rewrite the function as f(x) (x 2 3x 2) /2 apply the general power rule:

7 54 Chapter 2 Differentiation: Basic Concepts f(x) 2 (x2 3x 2) /2 d (x2 3x 2) 2 (x2 3x 2) /2 (2x 3) 2x 3 2x 2 3x 2 EXAMPLE 5.7 Differentiate the function f(x). (2x 3) 5 Do not use the quotient rule! It s much easier to rewrite the function as apply the general power rule to get f(x) (2x 3) 5 f(x) 5(2x 3) 6 d (2x 3) 5(2x 3)6 (2) 0 (2x 3) 6 The chain rule is often used in combination with the other rules you learned in Sections 2 3. The next example involves the product rule. EXAMPLE 5.8 Differentiate the function f(x) (3x ) 4 (2x ) 5 simplify your answer. First apply the product rule to get f(x) (3x ) 4 d [(2x )5 ] (2x ) 5 d [(3x )4 ] Continue by applying the general power rule to each term: f(x) (3x ) 4 [5(2x ) 4 (2)] (2x ) 5 [4(3x ) 3 (3)] 0(3x ) 4 (2x ) 4 2(2x ) 5 (3x ) 3 Finally, simplify your answer by factoring:

8 Chapter 2 Section 5 The Chain Rule 55 f(x) 2(3x ) 3 (2x ) 4 [5(3x ) 6(2x )] 2(3x ) 3 (2x ) 4 [5x 5 2x 6] 2(3x ) 3 (2x ) 4 (27x ) EXAMPLE 5.9 Differentiate the function f(t) First rewrite the function as t t then apply the general power rule to get f(t) 2 t t /2 Now use the quotient rule to find that substitute the result into the equation for f(t): simplify your answer. f(t) t t /2 d dt t t d dt t (t )() (t )() t (t ) 2 2 (t ) 2 f(t) 2 t t /2 t t /2 2 2 (t ) 2 (t ) Note that f() is defined ( f() 0), but f() is not. EXAMPLE 5.0 An environmental stu of a certain suburban community suggests that the average daily level of carbon monoxide in the air will be c(p) 0.5p 2 7 parts per million when the population is p thous. It is estimated that t years from now, the population of the community will be p(t) 3. 0.t 2 thous. At what rate will the carbon monoxide level be changing with respect to time 3 years from now?

9 56 Chapter 2 Differentiation: Basic Concepts Explore! Store the function C(x).5x 2 7 in your graphing utility. Use the numeric differentiation feature of your utility, if you have one, to calculate C(3). The goal is to find when t 3. Since it follows from the chain rule that When t 3, so dc dt dc dp 2 (0.5p2 7) /2 [0.5(2p)] 2 p(0.5p2 7) /2 dc dc dp dt dp dt 2 p(0.5p2 7) /2 0.pt (0.2t) 0.5p 2 7 dc 0.(4)(3) dt 0.5(4) 2 7 dp 0.2t dt p(3) 3. 0.(3) part per million per year 25 5 P. R. O. B. L. E. M. S 2.5 In Problems through 0, use the chain rule to compute the derivative simplify your answer.. y u 2, u 3x 2 2. y 2u 2 u 5, u x 2 3. y u, u x 2 2x 3 4. y u 2 2u 3, u x 5. y, u x 2 6. y, u 3x 2 5 u 2 u 7. y, u x y u 2 u 2, u u x 9. y, u x 2 0. y u 2, u u x

10 Chapter 2 Section 5 The Chain Rule 57 In Problems through 6, use the chain rule to compute the derivative for the given value of x.. y 3u 4 4u 5, u x 3 2x 5; x 2 2. y u 5 3u 2 6u 5, u x 2 ; x 3. y u, u x 2 2x 6; x 3 4. y 3u 2 6u 2, u ; x x y, u 3 ; x u x y, u x 3 2x 5; x 0 u In Problems 7 through 36, differentiate the given function simplify your answer. 7. f(x) (2x ) 4 8. f(x) 5x f(x) (x 5 4x 3 7) f(t) (3t 4 7t 2 9) f(t) 22. f(x) 5t 2 6t 2 (6x 2 5x ) g(x) 24. f(s) 4x 2 5s f(x) 26. f(x) ( x 2 ) 4 3(5x 4 ) h(s) ( 3s) g(x) 3x 29. f(x) (x 2) 3 (2x ) f(x) 2(3x ) 4 (5x 3) 2 3x 3. G(x) 2x 32. f(y) y 2 2 y 3 (x ) 5 ( 2x) f(x) 34. F(x) ( x) 4 (3x ) 3 3y 5x f(y) 36. f(x) 4y 3 3 2x In Problems 37 through 40, find an equation of the line that is tangent to the graph of f for the given value of x. 37. f(x) (3x 2 ) 2 ; x 38. f(x) (x 2 3) 5 (2x ) 3 ; x 2

11 58 Chapter 2 Differentiation: Basic Concepts 39. f(x) ; x 40. f(x) x ; x 3 x 3 (2x ) 6 In Problems 4 through 46, find all values of x where the tangent line to y f(x) is horizontal. 4. f(x) (x 2 x) f(x) x 3 (2x 2 x 3) 2 x 2x f(x) 44. f(x) (3x 2) 2 ( 2x) f(x) x 2 4x f(x) (x ) 2 (2x 3) 3 In Problems differentiate the given function f(x) by two different methods, first by using the general power rule then by using the product rule. Show that the two answers are the same. 47. f(x) (3x 5) f(x) (7 4x) 2 ANNUAL EARNINGS 49. The gross annual earnings of a certain company are f(t) 0t 2 t 236 thous dollars t years after its formation in January 998. (a) At what rate will the gross annual earnings of the company be growing in January 2003? (b) At what percentage rate will the gross annual earnings be growing in January 2003? MANUFACTURING COST CONSUMER DEMAND CONSUMER DEMAND CONSUMER DEMAND 50. At a certain factory, the total cost of manufacturing q units during the daily production run is C(q) 0.2q 2 q 900 dollars. From experience it has been determined that approximately q(t) t 2 00t units are manufactured during the first t hours of a production run. Compute the rate at which the total manufacturing cost is changing with respect to time hour after production commences. 5. When electric blenders are sold for p dollars apiece, local consumers will buy 8,000 D(p) blenders a month. It is estimated that t months from now, the p price of the blenders will be p(t) 0.04t 3/2 5 dollars. Compute the rate at which the monthly dem for the blenders will be changing with respect to time 25 months from now. Will the dem be increasing or decreasing? 52. An importer of Brazilian coffee estimates that local consumers will buy 4,374 approximately D( p) pounds of the coffee per week when the price is p p 2 dollars per pound. It is also estimated that t weeks from now, the price of Brazilian coffee will be p(t) 0.02t 2 0.t 6 dollars per pound. At what rate will the weekly dem for the coffee be changing with respect to time 0 weeks from now? Will the dem be increasing or decreasing? 53. When a certain commodity is sold for p dollars per unit, consumers will buy 40,000 D( p) units per month. It is estimated that t months from now, the price p

12 Chapter 2 Section 5 The Chain Rule 59 AIR POLLUTION ANIMAL BEHAVIOR of the commodity will be p(t) 0.4t 3/2 6.8 dollars per unit. At what percentage rate will the monthly dem for the commodity be changing with respect to time 4 months from now? 54. It is estimated that t years from now, the population of a certain suburban community will be p(t) 20 thous. An environmental stu indicates that 6 t the average daily level of carbon monoxide in the air will be c( p) 0.5p 2 p 58 parts per million when the population is p thous. Find the rate at which the level of carbon monoxide will be changing with respect to time 2 years from now. 55. In a research paper,* V. A. Tucker K. Schmidt-Koenig demonstrated that a species of Australian parakeet (the Budgerigar) expends energy according to the formula E v [0.074(v 35)2 32] COMPOUND INTEREST BLOOD FLOW where v is the bird s velocity (in km/hr). Find a formula for the rate of change of E with respect to velocity v. 56. If $0,000 is invested at an annual rate r (expressed as a decimal) compounded weekly, the total amount (principal P interest) accumulated after 0 years is given by the formula A 0,000 r (a) Find the rate of change of A with respect to r. (b) Find the percentage rate of change of A with respect to r when r 0.05 (that is, 5%). 57. It has been determined that the flow of blood from an artery into a small capillary is given by the formula F kd 2 A C (cm 3 /sec) where D is the diameter of the capillary, A is the pressure in the artery, C is the pressure in the capillary, k is a positive constant. (a) By how much is the flow of blood F changing with respect to pressure C in the capillary if A D are kept constant? Does the flow increase or decrease with increasing C? (b) What is the percentage rate of change of flow F with respect to A if C D are kept constant? * V. A. Tucker K. Schmidt-Koenig, Flight Speeds of Birds in Relation to Energetics Wind Directions, The Auk, Vol. 88 (97), pages

13 60 Chapter 2 Differentiation: Basic Concepts LEARNING CURVES 58. When you first begin to stu a topic or practice a skill, you may not be very good at it, but in time, you will approach the limits of your ability. One model for describing this behavior involves the equation T all b where T is the time required for a particular person to learn the items on a list of L items a b are positive constants. dt (a) Find the derivative interpret it in terms of the learning model. dl (b) Read discuss in one paragraph an article on how learning curves can be used to stu worker productivity.* 59. Suppose L(x) is a function with the property that L(x). Use the chain rule x to find the derivatives of the following functions simplify your answers. (a) f(x) L(x 2 ) (b) f(x) (c) f(x) (d) f(x) L x L 2 3x L 2x x 60. Prove the general power rule for n 2 by using the product rule to compute if y [h(x)] Prove the general power rule for n 3 by using the product rule the result of Problem 60 to compute if y [h(x)] 3. [Hint: Begin by writing y as h(x)[h(x)] 2.] 62. Store the function f(x) 3 3.x in your graphing utility. Use the numeric differentiation feature of your utility to calculate f() f(3). Explore the graph of f(x). How many horizontal tangents does the graph have? 63. Store the function f(x) (2.7x 3 3x 5) 2/3 in your graphing utility. Use the numeric differentiation feature of the utility to calculate f(0) f(4.3). Explore the graph of f(x). How many horizontal tangents does it have? * You may wish to begin your research by consulting Philip E. Hicks, Industrial Engineering Management: A New Perspective, 2nd ed., Chapter 6, McGraw-Hill, Inc., New York, 994, pages