The Derivative of ln x. d dx EXAMPLE 3.1. Differentiate the function f(x) x ln x. EXAMPLE 3.2

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1 Chapter 4 Section 3 Differentiation of Logarithmic an Eponential Functions RADIOLOGY Differentiation of Logarithmic an Eponential Functions 61. The raioactive isotope gallium-67 ( 67 Ga), use in the iagnosis of malignant tumors, has a half-life of 46.5 hours. If we start with 100 milligrams of the isotope, how many milligrams will be left after 24 hours? When will there be only 25 milligrams left? Answer these questions by first using the graphing utility to graph an appropriate eponential function an then using the trace an zoom features. Eponential functions an logarithmic functions both have simple erivatives. In this section, we will obtain erivative formulas for these functions an use them to analyze a few practical problems. We begin with the erivative formula for ln. The Derivative of ln (ln ) 1 for 0 The erivation of this formula is given at the en of this section, after we eamine a variety of eamples involving its use. EXAMPLE 3.1 Graph y ln using a moifie ecimal winow, [0.7, 8.7]1 by [3.1, 3.1]1. Choose a value of an construct the tangent line to the curve at this. Observe how close the slope of the 1 tangent line is to. Repeat this for several aitional values of. Differentiate the function f() ln. Combine the prouct rule with the formula for the erivative of ln to get EXAMPLE 3.2 ln 3 2 Differentiate f(). 4 f() 1 ln 1 ln First, since 3 2 2/3, the power rule for logarithms allows us to write 2 f() ln 3 2 ln 2/3 3 ln 4 4 4

2 332 Chapter 4 Eponential an Logarithmic Functions Then, by the quotient rule, we fin f() (ln ) ( 4 ) ln ( 4 ) ln ln 5 EXAMPLE 3.3 Differentiate g(t) (t ln t) 3/2. The function has the form g(t) u 3/2, where u t ln t, an by applying the general power rule, we fin g(t) t u3/2 3 2 u1/2 u t 3 2 (t ln t)1/2 (t ln t) t 3 2 (t ln t)1/2 1 1 t If f() ln u(), where u() is a ifferentiable function of, then the chain rule yiels the following formula for f(). The Chain Rule for Logarithmic Functions If u() is a ifferentiable function of, then 1 [ln u()] u() u

3 Chapter 4 Section 3 Differentiation of Logarithmic an Eponential Functions 333 EXAMPLE 3.4 Differentiate the function f() ln (2 3 1). Here, we have f() ln u, where u() Thus, f() 1 u u (32 ) (23 1) EXAMPLE 3.5 A manufacturer estimates that units of a particular commoity will be sol when the price is p() 112 ln 3 hunre ollars per unit. What is the marginal revenue associate with this commoity when 4 units are sol? The revenue is R() p() (112 ln 3 ) (3 ln ) hunre ollars, so the marginal revenue is R() When 4, the marginal revenue is (2) ln ln 3 R(4) 112 6(4) ln 4 3(4) hunre ollars per unit; that is, $6,673 per unit. EXPONENTIAL FUNCTIONS To fin the formula for the erivative of e, ifferentiate both sies of the equation ln e with respect to, using the chain rule for logarithms to ifferentiate ln e. You get (e ) e 1 or (e ) e

4 334 Chapter 4 Eponential an Logarithmic Functions Graph y e using a moifie ecimal winow, [0.7, 8.7]1 by [.1, 6.1]1. Trace the curve to any value of an etermine the value of the erivative at this point. Observe how close the erivative value is to the y coorinate of the graph. Repeat this for several values of. That is, e is its own erivative! Geometrically, this means that at each point P(c, e c ) on the curve y e, the slope is equal to e c, the y coorinate of P (see Figure 4.8). This feature is the main reason the number e is calle the natural eponential base. y y = e (c, e c ) The slope is e c FIGURE 4.8 At each point P(c, e c ) on the graph of y e, the slope equals e c. By using the chain rule in conjunction with the ifferentiation formula (e ) e we obtain the following formula for ifferentiating general eponential functions. The Chain Rule for Eponential Functions If u() is a ifferentiable function of, then (e u() u() u ) e EXAMPLE 3.6 Differentiate the function f(). e 2 1 By the chain rule, f() e 2 1 (2 1) 2e 2 1

5 Chapter 4 Section 3 Differentiation of Logarithmic an Eponential Functions 335 EXAMPLE 3.7 Differentiate the function f() e 2. By the prouct rule, f() (e2 ) e 2 () (2e2 ) e 2 (1) (2 1)e 2 EXAMPLE 3.8 Differentiate the function Using the quotient rule, you get e3 f() 2 1 f() (2 1)(3e 3 ) (2)e 3 ( 2 1) 2 e 3 3(2 1) 2 ( 2 1) 2 e ( 2 1) 2 Base on Eample 3.9, store the function y Ae B in Y1 of the equation eitor. For ifferent values of A an B fin the location of the maimum of y in terms of A an B. For eample, set A 1 an then vary the value of B (say, 1, 0.5, an 0.1) to see where the maimal functional value occurs. Then fi B at 0.1 an let A vary (say, 1, 10, 100). Make a conjecture about the location of the maimal y value in this case. EXAMPLE 3.9 A manufacturer estimates that D(p) 5,000e 0.02p units of a particular commoity will be sol when the price is p ollars per unit. Determine the market price p that will result in the largest revenue R pd(p). The revenue function is R(p) pd(p) 5,000pe 0.02p for p 0 (only nonnegative prices have economic meaning), with erivative R(p) 5,000(0.02pe 0.02p e 0.02p ) 5,000(1 0.02p)e 0.02p

6 336 Chapter 4 Eponential an Logarithmic Functions R(p) Maimum revenue 50 FIGURE 4.9 Revenue R 5,000pe 0.02p. p Since e 0.02p is always positive, R(p) 0 if an only if so p 0 or p To verify that p 50 actually gives the absolute maimum, note that R(p) 5,000(0.0004p 0.04)e 0.02p R(50) 5,000[0.0004(50) 0.04]e 0.02(50) 37 0 Thus, the secon erivative test tells you that the absolute maimum of R(p) oes inee occur when p 50 (see Figure 4.9). EXPONENTIAL GROWTH In Section 1, you saw that a quantity Q(t) that increases accoring to a law of the form Q(t) Q 0 e kt, where Q 0 an k are positive constants, is sai to eperience eponential growth. The net eample shows that if a quantity grows eponentially, its rate of growth is proportional to its size an its percentage rate of growth is constant. The converse of these facts will be establishe in Chapter 5. EXAMPLE 3.10 Suppose Q(t) grows eponentially. (a) Show that the rate of change of Q with respect to t is proportional to its size. (b) Show that the percentage rate of change of Q with respect to t is constant. If Q(t) grows eponentially, it is given by a function of the form Q(t) Q 0 e kt, where Q 0 an k are positive constants. (a) The rate of change of Q with respect to t is the erivative Q(t) kq 0 e kt kq(t) This says that the rate of change, Q(t), is proportional to Q(t) itself an that the constant k that appears in the eponent of Q(t) is the constant of proportionality. (b) The percentage rate of change of Q with respect to t is 100 Q(t) Q(t) 100 kq(t) Q(t) 100k This says that the constant k, which appears in the eponent of the function Q(t) Q 0 e kt, is the percentage rate of change of Q with respect to t (epresse as a ecimal).

7 Chapter 4 Section 3 Differentiation of Logarithmic an Eponential Functions 337 Note Observe that this is consistent with what you alreay know about compoun interest; namely, that if interest is compoune continuously, the balance after t years is B(t) Pe rt, where r is the interest rate epresse as a ecimal. Eponential Growth If Q(t) grows eponentially accoring to the law Q(t) Q 0 e kt, then, 1. The rate of change of Q with respect to t is proportional to its size. Rate of change Q(t) kq(t) 2. The percentage rate of change of Q with respect to t is constant. Percentage rate of change 100 Q(t) Q(t) 100k LOGARITHMIC DIFFERENTIATION Sometimes you can simplify the work involve in ifferentiating a function if you first take its logarithm. This technique, calle logarithmic ifferentiation, is illustrate in the following eample. EXAMPLE Differentiate the function f(). (1 3) 4 You coul o this problem using the quotient rule an the chain rule, but the resulting computation woul be somewhat teious. (Try it!) A more efficient approach is to take logarithms of both sies of the epression for f: ln f() ln 3 1 (1 3) 4 ln 3 ( 1) ln (1 3) 4 1 ln ( 1) 4 ln (1 3) 3 (Notice that by introucing the logarithm, you eliminate the quotient, the cube root, an the fourth power.) Now use the chain rule for logarithms to ifferentiate both sies of this equation to get

8 338 Chapter 4 Eponential an Logarithmic Functions so that f() f() ( 1) f() f() (1 3) THE PROOF THAT (ln ) 1 1 The proof that (ln ) is base on the fact that approaches e as n increases (or ecreases) without boun. To erive the formula for the erivative of f() ln, form the ifference quotient an rewrite it using the properties of logarithms as follows: f( h) f() h ln ( h) ln h 1 h ln h ln h 1/h ln 1 h 1/h 1 1 n n for fie 0 To fin the erivative of ln, let h approach zero in the simplifie ifference quotient. This will be easier to o if you first let n. Then, h an so As h approaches zero, n sign of h. Since f( h) f() h h h 1 n an ln 1 1 n n/ ln 1 1 n n 1/ increases or ecreases without boun, epening on the lim nfi 1 n 1 n e 1 h n

9 Chapter 4 Section 3 Differentiation of Logarithmic an Eponential Functions 339 it follows that f( h) f() (ln ) lim nfi 0 h lim nfi ln 1 1 n n 1/ ln lim nfi 1 1 n n 1/ ln e 1/ 1 as claime. P. R. O. B. L. E. M. S 4.3 In Problems 1 through 20 ifferentiate the given function. 1. f() e 5 2. f() 3e f() e f() e 1/ 5. f() 30 10e f() 2 e 7. f() ( 2 3 5)e 6 8. f() 9. f() (1 3e ) f() 11. f() 12. f() e 1/(2) e f() ln f() ln f() 2 ln 16. f() ln ln 17. f() 3 e f() 19. f() ln f() e ln 1 In Problems 21 through 26, fin an equation for the tangent line to y f() at the specifie point. 21. f() e ; where f() ( 1)e 2 ; where 0 ln 23. f() ; where f() ; where 1 2 e 2 e 2 1 e 25. f() 2 ln ; where f() ln ; where e

10 340 Chapter 4 Eponential an Logarithmic Functions In Problems 27 through 32, use logarithmic ifferentiation to fin f(). ( 2) f() f() f() ( 1) 3 (6 ) f() 31. f() 32. f() e (6 5) 4 MARGINAL ANALYSIS POPULATION GROWTH COMPOUND INTEREST DEPRECIATION NEWTON S LAW OF COOLING In Problems 33 through 36, the ollar cost C() of proucing units of a particular commoity is given, along with the price p() at which all units can be sol. In each case, fin (a) Marginal cost. C() (b) Average unit cost A() an marginal average cost. (c) Revenue R() p() an marginal revenue. () The level of prouction where marginal revenue equals marginal cost. (e) The level of prouction where marginal cost equals average cost. 33. C() e 0.2 ; p() e C() 3 20; p() 2e C() 2 2; p() ln ( 3) C() 9 5e 2 ; p() e 37. It is projecte that t years from now, the population of a certain country will become P(t) 50e 0.02t million. (a) At what rate will the population be changing with respect to time 10 years from now? (b) At what percentage rate will the population be changing with respect to time t years from now? Does this percentage rate epen on t or is it constant? 38. Money is eposite in a bank offering interest at an annual rate of 6% compoune continuously. Fin the percentage rate of change of the balance with respect to time. 39. A certain inustrial machine epreciates so that its value after t years becomes Q(t) 20,000e 0.4t ollars. (a) At what rate is the value of the machine changing with respect to time after 5 years? (b) At what percentage rate is the value of the machine changing with respect to time after t years? Does this percentage rate epen on t or is it constant? 40. A cool rink is remove from a refrigerator on a hot summer ay an place in a room whose temperature is 30 Celsius. Accoring to a law of physics, the

11 Chapter 4 Section 3 Differentiation of Logarithmic an Eponential Functions 341 MARGINAL ANALYSIS CONSUMER EXPENDITURE OZONE DEPLETION ECOLOGY ALCOHOL ABUSE CONTROL temperature of the rink t minutes later is given by a function of the form f(t) 30 Ae kt. Show that the rate of change of the temperature of the rink with respect to time is proportional to the ifference between the temperature of the room an that of the rink. 41. The mathematics eitor at a major publishing house estimates that if thousan complimentary copies are istribute to professors, the first-year sales of a certain new tet will be f() 20 15e 0.2 thousan copies. Currently, the eitor is planning to istribute 10,000 complimentary copies. (a) Use marginal analysis to estimate the increase in first-year sales that will result if 1,000 aitional complimentary copies are istribute. (b) Calculate the actual increase in first-year sales that will result from the istribution of the aitional 1,000 complimentary copies. Is the estimate in part (a) a goo one? 42. The consumer eman for a certain commoity is D(p) 3,000e 0.01p units per month when the market price is p ollars per unit. Epress consumers total monthly epeniture for the commoity as a function of p an etermine the market price that will result in the greatest consumer epeniture. 43. It is known that fluorocarbons have the effect of epleting ozone in the upper atmosphere. Suppose it is foun that the amount Q of ozone in the atmosphere is eplete by 0.15% per year, so that after t years, the amount of original ozone Q 0 that remains is Q Q 0 e t (a) How long will it take before half the original ozone is eplete? (b) How many years will it take for the epletion of ozone to be 80% complete? 44. In a moel evelope by John Helms,* the water evaporation E(T) for a ponerosa pine is given by E(T) 4.6e 17.3T/(T237) where T (egrees Celsius) is the surrouning air temperature. (a) What is the rate of evaporation when T 30 C? (b) What is the percentage rate of evaporation? At what temperature oes the percentage rate of evaporation first rop below 0.5? 45. Suppose the percentage of alcohol in the bloo t hours after consumption is given by C(t) 0.2te t/2 (a) What is the maimum level of alcohol in the bloo? When oes it occur? (b) How much time must pass before the bloo alcohol level is 30% of the maimum level? * John A. Helms, Environmental Control of Net Photosynthesis in Naturally Growing Pinus Ponerosa Nets, Ecology (Winter, 1972), page 92.

12 342 Chapter 4 Eponential an Logarithmic Functions DERIVATIVES OF b AND log b FOR BASE b fi e 46. Let b be a positive number other than 1 (b 0, b 1). (a) Show that [Hint: Use the fact that b e ln b ] (b) Show that (b ) (ln b)b (log b ) 1 1 (ln b) Hint: Use the conversion formula log b ln ln b In Problems 47 through 50, use the formulas in Problem 46 to ifferentiate the given function f() 48. f() log f() log f() 51. Show that the percentage rate of change of f with respect to is 100 [ln f()]. In Problems 52 an 53 use the formula from Problem 51 to fin the specifie percentage rate of change. POPULATION GROWTH 52. A quantity grows accoring to the law Q(t). Fin the percentage rate of t change of Q with respect to t. 53. It is projecte that years from now the population of a certain town will be approimately P() 5, At what percentage rate will the population be changing with respect to time 3 years from now? 54. Use a numerical ifferentiation utility to fin f(c), where c 0.65 an f() ln Q 0 ekt 3 1 (1 3) 4 Then use a graphing utility to sketch the graph of f() an to raw the tangent line at the point where c. 55. Repeat Problem 54 with the function f() ( )e 32 an c 2.17.