SECTION 9.5 EXAMPLE The Product Rule and the Quotient Rule 587 Sensitivity to a Drug | APPLICATION PREVIEW | When medicine is administered, reaction (measured in change of blood pressure or temperature) can be modeled by R ϭ m2 c m Ϫ where c is a positive constant and m is the amount of medicine absorbed into the blood The rate of change of R with respect to m is the sensitivity of the body to medicine Find an expression for sensitivity s as a function of m Solution The sensitivity is the rate of change of R with respect to m, or the derivative Thus sϭ ✓ CHECKPOINT ANSWERS dg df d ϩg ( fg) ϭ f dx dx dx f g f Ј Ϫ f gЈ d ϭ False The derivative of a quotient is dx g g2 (a) f Ј(x) ϭ (x 12 ϩ 8x Ϫ 7)(70x Ϫ 4) ϩ (10x Ϫ 4x ϩ 19)(12x 11 ϩ 40x ) Ϫ20x (b) f Ј(x) ϭ (3x ϩ 2)2 No; yЈ can be found with the Coefficient Rule: yЈ ϭ (2x ϩ 3) f Ј(x) No; y Ј can be found with the Coefficient Rule: yЈ ϭ c False The derivative of a product is | EXERCISES | c dR ϭ m2 Ϫ Ϫ m (2m) ϩ dm 3 ϭ Ϫ m2 ϩ mc Ϫ m2 ϭ mc Ϫ m2 3 9.5 In Problems 1–4, find the derivative and simplify y ϭ (5x ϩ 3)(x Ϫ 2x) s ϭ (t ϩ 1)(t Ϫ 1) f (x) ϭ (x 12 ϩ 3x ϩ 4)(2x Ϫ 1) y ϭ (3x ϩ 4)(8x Ϫ 6x Ϫ 9) In Problems 5–8, find the derivative, but not simplify your answer y ϭ (7x Ϫ 5x ϩ 2x Ϫ 1)(4x ϩ 3x Ϫ 5x ϩ 3x) y ϭ (9x Ϫ 7x Ϫ 6x)(3x Ϫ 4x ϩ 3x Ϫ 8) y ϭ (x ϩ x ϩ 1)( x Ϫ x ϩ 5) y ϭ ( x Ϫ x ϩ 1)(x Ϫ 5x Ϫ 7) In Problems and 10, at each indicated point find (a) the slope of the tangent line, and (b) the instantaneous rate of change of the function y ϭ (x ϩ 1)(x Ϫ 4x) at (Ϫ2, 0) 10 y ϭ (x Ϫ 3)(x Ϫ 4x ϩ 1) at (2, Ϫ15) In Problems 11–20, find the indicated derivatives and simplify q2 ϩ dp for p ϭ 11 dq 2q Ϫ 2x 12 CЈ(x) for C(x) ϭ 3x ϩ dy Ϫ 2x 13 for y ϭ dx x Ϫ 2x ϩ t3 Ϫ ds for s ϭ 14 dt t Ϫ 2t Ϫ t Ϫ dz x2 15 for z ϭ x ϩ dx Ϫ x Ϫ 2x dy 100x for y ϭ 200x Ϫ 16 dx 3x ϩ dp 3q 17 for p ϭ dq 1Ϫq Copyright 2016 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 ■ 588 CHAPTER dy Derivatives xϪ1 dx Ϫ x3 x(x ϩ 4) 19 yЈ for y ϭ xϪ2 (x ϩ 1)(x Ϫ 2) 20 f Ј(x) for f (x) ϭ x2 ϩ 18 for y ϭ In Problems 21 and 22, at the indicated point for each function, find (a) the slope of the tangent line, and (b) the instantaneous rate of change of the function x2 ϩ 21 y ϭ at (2, 1) xϩ3 x Ϫ 4x 22 y ϭ at 2, Ϫ x ϩ 2x In Problems 23–26, write the equation of the tangent line to the graph of the function at the indicated point Check the reasonableness of your answer by graphing both the function and the tangent line 23 y ϭ (9x Ϫ 6x ϩ 1)(1 ϩ 2x) at x ϭ 24 y ϭ (4x ϩ 4x ϩ 1)(7 Ϫ 2x) at x ϭ 3x Ϫ 2x Ϫ 25 y ϭ at x ϭ Ϫ x2 x Ϫ 4x at x ϭ 26 y ϭ 2x Ϫ x In Problems 27–30, use the numerical derivative feature of a graphing calculator to find the derivative of each function at the given x-value 3 x Ϫ Ϫ 25 at x ϭ 27 y ϭ x ϩ x x at x ϭ 28 y ϭ (3 x ϩ x Ϫ 1) Ϫ x x 4x Ϫ at x ϭ 29 f (x) ϭ 3x 3 3xϩ1 30 f (x) ϭ at x ϭ Ϫ1 xϩ2 In Problems 31–34, complete the following (a) Find the derivative of each function, and check your work by graphing both your calculated derivative and the numerical derivative of the function (b) Use your graph of the derivative to find points where the original function has horizontal tangent lines (c) Use a graphing calculator to graph the function and indicate the points found in part (b) on the graph 31 f (x) ϭ (x ϩ 4x ϩ 4)(x Ϫ 7) 32 f (x) ϭ (x Ϫ 14x ϩ 49)(2x ϩ 1) x2 x2 Ϫ 33 y ϭ 34 y ϭ xϪ2 4Ϫx In Problems 35 and 36, (a) find f (x) (b) graph both f (x) and f (x) with a graphing utility (c) identify the x-values where f (x) ؍0, f (x) 0, and f (x) (d) identify x-values where f (x) has a maximum point or a minimum point, where f (x) is increasing, and where f (x) is decreasing 10x Ϫ x2 35 f (x) ϭ 36 f (x) ϭ x ϩ1 x ϩ4 37 Prove the Quotient Rule for differentiation (Hint: Add [Ϫu(x) v(x) ϩ u(x) v(x)] to the expanded numerator and use steps similar to those used to prove the Product Rule.) 38 Use the Quotient Rule to show that the Powers of x Rule applies to negative integer powers That is, show that (d dx)x n ϭ nx nϪ1 when n ϭ Ϫk, k Ͼ 0, by finding the derivative of f (x) ϭ (x k) APPLICATIONS 39 Cost-benefit If the cost C (in dollars) of removing p percent of the particulate pollution from the exhaust gases at an industrial site is given by C(p) ϭ 8100p 100 Ϫ p find the rate of change of C with respect to p 40 Cost-benefit If the cost C (in dollars) of removing p percent of the impurities from the waste water in a manufacturing process is given by C( p) ϭ 9800p 101 Ϫ p find the rate of change of C with respect to p 41 Revenue Suppose the revenue (in dollars) from the sale of x units of a product is given by R(x) ϭ 60x ϩ 74x 2x ϩ Find the marginal revenue when 49 units are sold Interpret your result 42 Revenue The revenue (in dollars) from the sale of x units of a product is given by R(x) ϭ 3000 ϩ 80x Ϫ 1500 2x ϩ Find the marginal revenue when 149 units are sold Interpret your result 43 Revenue A travel agency will plan a group tour for groups of size 25 or larger If the group contains exactly 25 people, the cost is $300 per person If each person’s cost is reduced by $10 for each additional person above the 25, then the revenue is given by R(x) ϭ (25 ϩ x)(300 Ϫ 10x) Copyright 2016 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 SECTION 9.5 where x is the number of additional people above 25 Find the marginal revenue if the group contains 30 people Interpret your result 44 Revenue McRobert’s Electronics sells 200 TVs per month at a price of $400 per unit Market research indicates that the store can sell one additional TV for each $1 it reduces the price In this case the total revenue is R(x) ϭ (200 ϩ x)(400 Ϫ x) where x is the number of additional TVs beyond the 200 If the store sells a total of 250 TVs, find the marginal revenue Interpret your result 45 Response to a drug The reaction R to an injection of a drug is related to the dosage x (in milligrams) according to R(x) ϭ x 500 Ϫ N(t) ϭ 25t ϩ Ϫ2 t2 ϩ Find the rate at which the action potentials are produced by the nerve 47 Test reliability If a test having reliability r is lengthened by a factor n, the reliability of the new test is given by Rϭ nr , 0ϽrՅ1 ϩ (n Ϫ 1)r Find the rate at which R changes with respect to n 48 Advertising and sales The sales of a product s (in thousands of dollars) are related to advertising expenses (in thousands of dollars) by sϭ 200x x ϩ 10 Find and interpret the meaning of the rate of change of sales with respect to advertising expenses when (a) x ϭ 10 (b) x ϭ 20 49 Candidate recognition Suppose that the proportion P of voters who recognize a candidate’s name t months after the start of the campaign is given by 13t P(t) ϭ ϩ 0.18 t ϩ 100 (a) Find the rate of change of P when t ϭ 6, and explain its meaning (b) Find the rate of change of P when t ϭ 12, and explain its meaning (c) One month prior to the election, is it better for PЈ(t) to be positive or negative? Explain 589 50 Endangered species population It is determined that a wildlife refuge can support a group of up to 120 of a certain endangered species If 75 are introduced onto the refuge and their population after t years is given by p(t) ϭ 75 ϩ 4t t ϩ 16 find the rate of population growth after t years Find the rate after each of the first years 51 Wind chill In January 2014 the so-called “Polar Vortex” of dense, frigid air plunged deep into the United States and resulted in record cold temperatures and dangerous wind chills If s is the wind speed in miles per hour and s Ն 5, then the wind chill (in degrees Fahrenheit) for an air temperature of 0ЊF can be approximated by the function x where 1000 mg is the maximum dosage If the rate of reaction with respect to the dosage defines the sensitivity to the drug, find the sensitivity 46 Nerve response The number of action potentials produced by a nerve, t seconds after a stimulus, is given by The Product Rule and the Quotient Rule f (s) ϭ 289.173 Ϫ 58.5731s sϩ1 (a) At what rate is the wind chill changing when the wind speed is 20 mph? (b) Explain the meaning of your answer to part (a) 52 Response to injected adrenalin Experimental evidence has shown that the response y of a muscle is related to the concentration of injected adrenaline x according to the equation yϭ x a ϩ bx where a and b are constants Find the rate of change of response with respect to the concentration 53 Social Security beneficiaries The table gives the number of millions of Social Security beneficiaries (actual and projected) for selected years from 1950 through 2030 Year Number of Beneficiaries (millions) 1950 1960 1970 1980 1990 2.9 14.3 25.2 35.1 39.5 Year Number of Beneficiaries (millions) 2000 2010 2020 2030 44.8 53.3 68.8 82.7 Source: Social Security Trustees Report With B(t) representing the number of beneficiaries (in millions) t years past 1950, these data can be modeled by the function B(t) ϭ (0.01t ϩ 3)(0.0238t Ϫ 9.79t ϩ 3100)Ϫ 9290 (a) Find the function that gives the instantaneous rate of change of the number of beneficiaries (b) Find and interpret the instantaneous rate of change in 2020 (c) Use the data to determine which of the average rates of change (from 2010 to 2020, from 2020 to 2030, or from 2010 to 2030) best approximates the instantaneous rate from part (b) Copyright 2016 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 ... both f (x) and f (x) with a graphing utility (c) identify the x-values where f (x) ؍0, f (x) 0, and f (x) (d) identify x-values where f (x) has a maximum point or a minimum point, where... derivative feature of a graphing calculator to find the derivative of each function at the given x-value 3 x Ϫ Ϫ 25 at x ϭ 27 y ϭ x ϩ x x at x ϭ 28 y ϭ (3 x ϩ x Ϫ 1) Ϫ x x 4x Ϫ at x ϭ 29 f (x) ϭ... dx)x n ϭ nx nϪ1 when n ϭ Ϫk, k Ͼ 0, by finding the derivative of f (x) ϭ (x k) APPLICATIONS 39 Cost-benefit If the cost C (in dollars) of removing p percent of the particulate pollution from the