-0 0

t D(t)

1994 414.0

1996 469.5

1998 467.3

2000 456.4

2002 442.3

D(t) - D(1998)

t t - 1998

1994 13.3

1996 -1.1

2000 -5.5

2002 -6.3

• A NOTE ON UNITS

Theunits for theaverage rate ofchange !:iD/!:it are theunits for!:iDdivided by the units for!:it, namely, billions of dollars peryear.The instan- taneous rateofchange isthelimit of the aver- age rates of change, soit ismeasured in the same units: billions ofdollars peryear.

In the following example we estimate the rate of change of the national debt with respect to time. Here the function isdefined not by a formula but by a table of values.

EXAMPLE 7 Let D(t) be the Canadian gross public debt at time t.The table in the margin gives approximate values of this function by providing midyear estimates, in billions of dollars, from 1994to 2002. Interpret and estimate the value of D'(l998).

SOLUTION The derivative D'(l998) means the rate of change ofDwith respect totwhen t= 1998, that is, the rate of increase ofthe national debt in 1998.

According to Equation 5,

D'(l998) = lim

D(t) -

D(l998)

Hl998 t - 1998

So we compute and tabulate values of the difference quotient (the average rates of change) as shown in the table at the left. From this table we see that D'(l998) lies some- where between -1.1 and - 5,5 billion dollars per year. [Here we are making the reason- able assumption that the debt didn't fluctuate wildly between 1998 and 2002.] We estimate that the rate ofchange of the Canadian debt in 1998 was the average of these two numbers, namely

The minus sign means that the debt was decreasing at that time.

Another method would betoplot the debt function and estimate the slope of the tan-

gent line when t= 1998. 0

In Examples 3, 6, and 7 we saw three specific examples of rates of change: the veloci- tyof an object isthe rate of change of displacement with respect totime; marginal cost is the rate of change of production cost with respect to the number of items produced; the rate of change ofthe debt with respect to time isof interest in economics. Here is a small sample ofother rates ofchange: In physics, the rate ofchange of work with respect to time is called power. Chemists who study a chemical reaction are interested in the rate of change in theconcentration of a reactant with respect to time (called the rate of reaction).

A biologist isinterested in the rate ofchange of the population of a colony of bacteria with respect totime. In fact, the computation of rates of change is important in all of the natu- ralsciences, inengineering, and even in the social sciences. Further examples will be given in Section 3.7.

All these rates of change are derivatives and can therefore be interpreted as slopes of tangents. This gives added significance tothe solution of the tangent problem. Whenever we solve a problem involving tangent lines, we are not just solving a problem in geome- try.Weare also implicitly solving a great variety of problems involving rates of change in science and engineering.

-0

^EXERCISES

I. Acurve has equation y =f(x).

(a) Write an expression for the slope ofthe secant line through the points P(3,f(3» and Q(x,f(x».

(b) Write an expression for the slope ofthe tangent line atP.

ffi

^{2. Graph the curve y =sinxin the viewing} rectangles [-2,2]

by[-2,2], [ -I, 1] by[ -1, 1], and [-0.5,0.5] by [-0.5,0.5]'

What do you notice about the curve as you zoom in toward the origin?

3. (a) Find the slope of the tangent line to the parabola y =4x - x2 atthe point (1, 3)

(i) using Definition 1 (ii) using Equation 2 (b) Find anequation ofthe tangent line in part (a).

(c) Graph the parabola and the tangent line. As acheck on your work, zoom in toward the point (1, 3)until the parabola andthe tangent line are indistinguishable.

4. (a) Find the slope of the tangent line tothe curve y =x - x3

at the point (1,0)

(i) using Definition 1 (ii) using Equation 2 (b) Find an equation of the tangent line in part (a).

(c) Graph the curve and the tangent line in successively smaller viewing rectangles centered at(1, 0) until the curve and the line appear to coincide.

5-8 Find anequation ofthe tangent line to the curve at the given point.

~;]y=--,

x-I

(3,2) 6. y=2x3-5x, (-1,3)

x-2

2x

8. Y= ( )2' (0, 0)

x

+

1

[!](a) Find the slope of the tangent tothecurve y

=

3

+

4x2 - 2x3 at the point where x

=

a.

(b) Find equations of the tangent lines at the points (1,5) and (2,3).

(c) Graph the curve and both tangents on acommon screen.

10. (a) Find the slope ofthe tangent to the curve y ⁼

1/.);

at the point where x = a.

(b) Find equations of the tangent lines at the points (1, 1) and (4,

D-

(c) Graph the curve and both tangents on acommon screen.

II. (a) A particle starts bymoving to the right along a horizontal line; the graph of itsposition function is shown. When is the particle moving to the right? Moving tothe left?

Standing still?

(b) Draw agraph ofthe velocity function.

ers) 4

2 /

\

/

\

/

0 2 4 6 t

12. Shown are graphs ofthe position functions of two runners, A and B, who run a 100-m race and finish inatie.

s (meters)

80 /j

~ /

40

,/

/

/' /"8

IL

V ...-

0 4 8 12 t

(b) At what time isthe distance between the runners the greatest?

(c) At what time dothey have the same velocity?

13. If a ball isthrown into the airwith avelocity of 10 mis, its height (in meters) after t seconds isgiven by y = lOt - 4.9t2•

Find the velocity when t⁼2.

14. If a rock isthrown upward on the planet Mars with a velocity of 10 mis, itsheight (in meters) after t seconds is given by H= lOt - 1.86t2•

(a) Find the velocity ofthe rock after one second.

(b) Find the velocity ofthe rock when t=a.

(c) When willthe rock hit thesurface?

(d) With what velocity will the rock hit the surface?

15. The displacement (in meters) of aparticle moving ina straight line is given bythe equation of motion s⁼ l/t2,

where tis measured in seconds. Find the velocity ofthe particle at times t=a,t =1,t =2,and t=3.

16. The displacement (in meters) of aparticle moving in a straight line is given by s⁼t2 - 8t

+

18,where tismea- sured inseconds.

(a) Find the average velocity over each time interval:

(i) [3,4] (ii) [3.5,4]

(iii) [4,5] (iv) [4,4.5]

(b) Find the instantaneous velocity when t =4.

(c) Draw the graph ofs asa function of tand draw the secant lines whose slopes are theaverage velocities in part (a) and the tangent line whose slope isthe instantaneous velocity in part (b).

ITLl

^For the function 9whose graph isgiven, arrange the follow- ing numbers in increasing order and explain your reasoning:

1m

(a) Find an equation of the tangent line tothe graph of y =g(x) atx =5 if g(5) = -3 and g'(5) =4.

(b)If thetangent line to y =f(x) at(4,3) passes through the point (0, 2), find f(4) and1'(4).

[!!J

^Sketch the graph of a function f for which f(O) =0, 1'(0) =3,1'(1) =0,and 1'(2) = -1.

20. Sketch the graph ofa function 9 forwhich g(O)

=

g'(O)

=

0, g'(-1) = -1, g'(1) =3,and g'(2) = 1.

21. Iff(x) ⁼3x2 - 5x,find1'(2) and use it to find an equation of the tangent line to the parabola y =3x2 - 5x at the point (2,2).

22. Ifg(x)

=

1 - x3, find g'(O) and use it to find an equation of the tangent line to the curve y =1- x3 at the point (0, 1).

~ (a) If F(x) ⁼5x/(t

+

x2), find F'(2) and use it tofind an equation of the tangent line tothe curve y =5x/(t

+

x2)

atthe point (2, 2).

~ (b) Illustrate part (a) by graphing thecurve and the tangent line on the same screen.

24. (a) If G(x) ⁼4x2 - x3, find G'(a) and use it to find equa- tions of the tangent lines to the curve y =4x2 - x3 at the points (2,8)and (3,9).

~ (b) Illustrate part (a) bygraphing the curve and the tangent lines onthe same screen.

25-30 Find f'(a).

25. f(x) = 3- 2x

+

4x2

2t

+

1

lmf(t)

=--

t

+

3

26. f(t) =t4 - 5t x2

+

1 28. f(x) =--

x-2 1

29. f(x) = ~ yX

+

2

31-36 Each limit represents the derivative ofsome function

f

at some number a.State such an

f

and ain each case.

(l

+

h)'O - 1 31. Iim---

,,~O h

32. lim ---~-2

,,~O h

2' - 32 33.lim---

,~5 x - 5

tanx - 1 34. Hm ----

,~",/4 x - 71/4

I'>Fl cas(7T

+

h)

+

1

~lim---

,,~O h

t4

+

t - 2 36.lim----

1~1 t - 1

37-38 A particle moves along astraight linewith equation of motion s⁼f(t), where sismeasured in meters and t inseconds.

Find the velocity and thespeed when t =5.

m

A warm can ofsoda pop isplaced in a cold refrigerator.

Sketch the graph ofthetemperature ofthe soda asa function of time. Isthe initial rate ofchange oftemperature greater or less than the rate of change after an hour?

40. Aroast turkey istaken from anoven when itstemperature hasreached 85°C and isplaced on a table in a room where the temperature is24°C. The graph shows how the tempera- ture ofthe turkey decreases and eventually approaches room

temperature. (In Section 7.5 we will be able to use Newton's Law of Cooling to find anequation for T as a function of time.) By measuring the slope of the tangent, estimate the rate of change ofthe temperature after an hour.

(0e) 80

~

•••...P 40

-...

~

r--

0 30 60 90 120 150 t

41. The table shows the estimated percentage Pof the population of Europe that use cell phones. (Midyear estimates are given.)

Year 1998 1999 2000 2001 2002 2003

P 28 39 55 68 77 83

(a) Find theaverage rate of cell phone growth

(i) from 2000 to2002 (ii) from 2000 to 200 1 (iii) from 1999 to2000

In each case, include the units.

(b) Estimate the instantaneous rate of growth in 2000 by taking theaverage of two average rates of change. What are itsunits?

(c) Estimate the instantaneous rate of growth in 2000 bymea- suring the slope of a tangent.

42. The number N oflocations ofapopular coffeehouse chain is given in the table. (The numbers of locations as of June 30 are given.)

Year 1998 1999 2000 2001 2002

N 1886 2135 3501 4709 5886

(a) Find the average rate of growth

(i) from 2000 to 2002 (ii) from 2000 to 2001 (iii) from 1999 to2000

Ineach case, include the units.

(b) Estimate the instantaneous rate of growth in 2000 by taking the average of two average rates of change. What areitsunits?

(c) Estimate the instantaneous rate of growth in 2000 by mea- suring the slope ofa tangent.

I1!l

The cost (in dollars) ofproducing xunits of a certain com- modity is C(x) ⁼5000

+

lOx

+

0.05x2•

(a) Find the average rate ofchange of C with respect to x when the production level ischanged

(i) from x = 100 tox = 105 (ii) from x = 100 to x= 101

(b) Find the instantaneous rate of change of C with respect to x when x = 100. (This iscalled the marginal cost.

Its significance will be explained in Section 3.7.)

Ifacylindrical tank holds 100,000 liters of water, which can be drained from the bottom ofthe tank inan hour, then Torri- celli's Law gives the volume Vof water remaining in the tank after tminutes as

Vet)= 100,000 ( 1- ;0

Y

Find the rate at which the water isflowing out of the tank (the instantaneous rate of change of Vwith respect tot) asafunc- tion of t.What are itsunits? For times t=0, 10,20, 30,40, 50, and 60 min, find the flow rate and the amount of water remaining in the tank. Summarize your findings in asentence or two. At what time isthe flowrate thegreatest? The least?

45. The cost of producing xkilograms of gold from a new gold mine isC ⁼f(x) dollars.

(a) What is themeaning of the derivative 1'(x)?What are its units?

(b) What does the statement 1'(50) ⁼36 mean?

(c) Doyou think the values of1'(x) will increase or decrease in the short term? What about thelong term? Explain.

46. The number ofbacteria after thours in a controlled labora- tory experiment isn⁼f(t).

(a) What is the meaning ofthe derivative 1'(5)? What are its units?

(b) Suppose there isan unlimited amount ofspace and nutri- ents for the bacteria. Which doyou think islarger, 1'(5) or 1'(IO)? If the supply ofnutrients islimited, would that affect your conclusion? Explain.

47. Let T(t) be the temperature (in °C) in Seoul thours after noon on August 21,2004. The table shows values ofthis function recorded every two hours. What is the meaning ofT'(6)?

Estimate itsvalue.

t 0 2 4 6 8 10

T 34.4 35.6 38.3 32.8 26.1 22.8

48. The quantity (in kilograms) ofa gourmet ground coffee that is sold by a coffee company at a price of p dollars per kilogram is Q=f(p).

(a) What is themeaning of the derivative 1'(8)? What are its units?

(b) Is 1'(8) positive or negative? Explain.

49. The quantity of oxygen that can dissolve in water depends on thetemperature ofthe water. (So thermal pollution influences

the oxygen content of water.) The graph shows how oxygen solubility Svaries asa function of the water temperature T.

(a) What is the meaning ofthe derivative S'(T)? What are its units?

(b) Estimate the value ofS'(l6) and interpret it.

s

(mg/L) 16

AdaptedfromEnvironmental Science: Living Within the System of Nature. 2d ed..byCharles E.Kupchella. ©1989.Reprinted by permission of Prentice-Hall,Inc.. Upper SaddleRiver.NJ.

The graph shows the influence of thetemperature T on the maximum sustainable swimming speed S of Coho salmon.

(a) What is the meaning of the derivative S'(T)? What are its units?

(b) Estimate the values ofS'(l5) and S'(25) and interpret them.

s

(em/s) 20

{ . J

X SIn-

[ill

^{f(x) = 0 x}

{ . 1

X2SIn-

52. f(x) = 0 x

for all values of

x.

So fill is a constant function and its graph is a horizontal line. There- fore, for all values of x,

We can interpret the third derivative physically in the case where the function is the position function s =

set)

of an object that moves along a straight line. Because

Sill =(s")' =a', thethird derivative ofthe position function isthe derivative of theaccel- eration function and is called thejerk:

. da d3s

} = --:it ⁼

^dt3

Thus thejerkj isthe rate of change of acceleration. It is aptly named because a largejerk means a sudden change in acceleration, which causes an abrupt movement in avehicle.

Wehave seen that one application of second and third derivatives occurs inanalyzing the motion of objects using acceleration and jerk. We will investigate another applica- tion of second derivatives in Section 4.3, where we show how knowledge of f" gives us information about the shape of the graph of

f.

In Chapter 12we will see how second and higher derivatives enable ustorepresent functions as sums of infinite series.

@

^EXERCISES

1-2 Use the given graph toestimate the value of each derivative.

Then sketch the graph of

f'. rn

^{Match the graph ofeach function in(a)-(d)} with the graph of

itsderivative in I-IV. Give reasons for your choices.

I. (a) 1'(-3) (c)1'( -1) (e)1'(1) (g) 1'(3)

2. (a)1'(0) (c)1'(2) (e) 1'(4)

(b) 1'(-2) (d) 1'(0) (f) 1'(2)

(b) 1'(1) (d) 1'(3) (f) 1'(5)

4-11 Trace orcopy thegraph of the given function

f.

(Assume that the axes have equal scales.) Then use the method ofExample 1 tosketch the graph of

l'

below it.

4.

12. Shown isthe graph of the population function P(t) foryeast cells inalaboratory culture. Use the method ofExample 1 to graph the derivative P'(t). What does the graph of P' tell us about the yeast population?

13. The graph shows how the average age of first marriage of Japanese men has varied inthe last half ofthe 20th century.

Sketch the graph of the derivative function M'(t). During which years was the derivative negative?

-t

^{l 1960I 1970I 1980I 1990I 2000I •t}

14. Make acareful sketch ofthe graph of the sine function and below it sketch the graph of itsderivative in the same manner as in Exercises 4-11.Can you guess what the derivative of the sine function is from itsgraph?

ffi ~

^{Let f(x) =x2•}

(a) Estimate the values off'(O),

I'm,

1'(1), and1'(2) by using agraphing device to zoom inon the graph off (b) Use symmetry todeduce the values off'(

-0,1'(

-I),

andf'(-2).

(c) Use the results from parts (a) and (b) toguess a formula for f'(x).

(d) Use the definition of aderivative toprove that your guess in part (c)is correct.

ffi

^{16. Letf(x) =x3.}

(a) Estimate the values of 1'(0),

I'm,

1'(1), 1'(2), andf'(3) byusing agraphing device to zoom inon thegraph off (b) Use symmetry todeduce the values off'(

-n

^1'(-1),

1'(-2), and1'(-3).

(c) Use thevalues from parts (a) and (b)to graph

1'.

(d) Guess aformula for f'(x).

(e) Use the definition ofaderivative to prove that your guess in part (d) is correct.

17-27 Find the derivative of the function using the definition of derivative. State the domain of thefunction and the domain ofits derivative.

17. f(x) =~x - ~ 19. f(t) =5t - 9t2

21. f(x) =x3 - 3x

+

5

18.f(x) =mx

+

b 20. f(x) = 1.5x2 - X

+

3.7 22. f(x) =x

+ j;

24. f(x) =--3+x 1 - 3x

lID

G(t) =--4t t

+

1

1 26. g(t) ⁼

Jt

28. (a) Sketch the graph off(x) = ~ by starting with the 35.

graph ofy

= j";

and using the transformations of Sec- tion 1.3.

(b) Use the graph from part (a) tosketch the graph of f'.

(c) Use thedefinition of aderivative tofind1'(x). What are the domains of

f

and 1'?

~ (d) Use agraphing device to graph

l'

and compare with your sketch in part (b).

29. (a) If f(x) =x4

+

2x, find1'(x).

~ (b) Check tosee that your answer topart (a) isreasonable by comparing the graphs of

f

and f'.

30. (a) Iff(t) =t2 -

JI,

find1'(t).

~ (b) Check to see that your answer topart (a) is reasonable by comparing the graphs of

f

and f'.

31. The unemployment rate Vet)varies with time. The table gives the percentage of unemployed in the Australian labor force measured at midyear from 1995 to 2004.

t Vet) t Vet)

1995 8.1 2000 6.2

1996 8.0 2001 6.9

1997 8.2 2002 6.5

1998 7.9 2003 6.2

1999 6.7 2004 5.6

(a) What isthe meaning of V'et)? What are its units?

(b) Construct a table of values for V'et).

32. Letpet) be the percentage ofthe population ofthe Phillipines over the age of60at time t. The table gives projections of values of this function from 1995 to 2020.

t pet) t pet)

1995 5.2 2010 6.7

2000 5.5 2015 7.7

2005 6.1 2020 8.9

(a) What isthe meaning of P'(t)? What are itsunits?

(b) Construct atable ofestimated values for P'(t).

(c) Graph P and P'.

33-36 The graph of

f

is given. State, with reasons, the numbers at which

f

isnot differentiable.

~ 37. Graph the functionf(x) =x

+ M.

Zoom in repeatedly, first toward the point (-1, 0) and then toward the origin.

What isdifferent about the behavior of

f

in the vicinity of these two points? What do you conclude about the differen- tiabilityoff?

~ 38. Zoom intoward the points (1, 0), (0, I), and ( - 1,0) on the graph of the function g(x) = (x2 - 1?/3.What do you notice? Account for what you see in terms ofthe differen- tiability ofg.

~ The figure shows the graphs of

f, 1',

and

1".

Identify each curve, and explain your choices.

40. The figure shows graphs of f,

1',

f", and

1"'.

Identify each curve, and explain your choices.

41. The figure shows the graphs of three functions. One isthe position function of a car,one isthe velocity of the car, and one isitsacceleration. Identify each curve, and explain your choices.

42. The figure shows the graphs of four functions. One isthe position function of a car, one isthe velocity of the car, one is its acceleration, and one isitsjerk. Identify each curve, and explain your choices.

~ 43-44 Use the definition of aderivative to findf'(x) and

rCx).

Then graph f,

1',

and

r

^{on a}common screen and check tosee if your answers arereasonable.

~~ If f(x) =2x2 - x3, findf'(x), r(x),flll(x), and f(4)(X).

Graph f,

1',

f", and fill onacommon screen. Are the graphs consistent with the geometric interpretations ofthese derivatives?

46. (a) The graph of a position function of acar is shown, where sis measured in meters and t in seconds. Use it to graph the velocity and acceleration of the car.What is the accel- eration at t=10seconds?

(b) Use the acceleration curve from part (a)to estimate the jerk at t = 10seconds. What are the units for jerk?

47. Letf(x) ⁼

$.

(a) Ifa ¥- 0, use Equation 3.1.5 to findf'(a).

(b) Show thatf'(O) does not exist.

(c) Show thaty =

$

has avertical tangent line at(0,0).

(Recall the shape ofthe graph of

f.

SeeFigure 13inSec- tion 1.2.)

48. (a) If g(x) ⁼X2/3, show that g'(O) does not exist.

(b) If a¥- 0,findg'(a).

(c) Show that y ⁼X2/3 has a vertical tangent line at (0,0).

~ (d) Illustrate part (c) bygraphing y =X2/3•

lID

Show that the function f(x) =

I

x -

61

isnot differentiable at 6.Find a formula for

I'

and sketch itsgraph.

50. Where isthegreatest integer function f(x) =[x ~not differ- entiable? Find a formula for

I'

and sketch its graph.

[!.J

^(a) Sketch the graph ofthe function f(x) =x

I

x

I·

(b) Forwhat values ofx is

f

differentiable?

(c) Find a formula for

1'.

52. The left-hand and right-hand derivatives of

f

ata are defined by

f'-(a) = Jim f(a

+

h)- f(a)

h~O- h

f' ( ) - I' f(a

+

h) - f(a)

+ a - 1m

h~O+ h

ifthese limits exist. Then f'(a) exists ifand only ifthese one- sided derivatives exist and are equal.

(a) Find f'-(4) andf~(4) for the function

1

⁰

5-x

f(x) = _I_

s-x

if x ~ 0

if 0

<

x

<

4

(b) Sketch the graph of

f.

(c) Where is

f

discontinuous?

(d) Where is

f

not differentiable?

53. Recall that a function f is called even if f( -x) =f(x) for all x in its domain and odd if f( -x) =-f(x) for all such x.

Prove each of the following.

(a) The derivative ofan even function isan odd function.

(b) The derivative ofan odd function is aneven function.

54. When you turn on ahot-water faucet, the temperature Tof the water depends on how long the water has been running.

(a) Sketch apossible graph of T as a function ofthe time t that haselapsed since the faucet wasturned on.

(b) Describe how the rate of change of T with respect tot varies as t increases.

(c) Sketch a graph of the derivative ofT.

55. Let

e

^be the tangent line to the parabola y =x2 at the point (I, I).The angle of inclination of

e

^{isthe angle 4>that}

e

makes with the positive direction ofthe x-axis. Calculate 4>

correct tothe nearest degree.

TABLE OF

DIFFERENTIATION FORMULAS

§

^EXERCISES

1-20 Differentiate the function.

I. f(x) ⁼ 186.5 3. f(x) =5x - 1 5. f(x) ⁼x3 - 4x

+

6

1-3'12 1

20

^(l

⁺

¹²⁾² = --⁴

We use the point-slope form to write an equation ofthe tangent line at (1,

D:

The slope of the normal line at(1,

D

^isthe negative reciprocal of

-i,

namely 4, so an

x equation is

EXAMPLE 13 At what points on the hyperbola xy = 12 isthe tangent line parallel tothe line 3x

+

y =O?

SOLUTION Since xy =12 can be written asy = l2/x, we have

dy d 12

- =

12-

(X-I) =

12(-x-

²⁾ = --

dx dx x2

Let the x-coordinate of one ofthe points inquestion be

a.

Then the slope of the tangent line atthat point is -12/ a2. This tangent line will be parallel tothe line 3x

+

^{y =0, or}

y = -3x, if ithas the same slope, that is, -3. Equating slopes, we get

--=12 -3 a2

Therefore the required points are (2,6) and (-2, -6). The hyperbola and the tangents

are shown inFigure 6.

0

-d (c) =0

dx

-d (x") =nx,,-I dx

gl' - fg' g2

8. f(t) =it6 - 3t4

+

t

10. R(t) =5t-3/5

4. F(x) ⁼-4x 10

6. h(x) =(x - 2)(2x

+

3) _{12. R(x)}

JW

=-7- X

1 14.f(t) =

Jt - Jt

16. R(x) =-7-

JiG

X

18. g(u) =

J2

^u

+ J3;;

20. v ^{= (}

JX + ~ y

21. Find the derivative of y =(x2 + 1)(x3 + 1) in two ways: by using the Product Rule and by performing the multiplication first. Do your answers agree?

() x - 3xJX

Fx=

JX

in two ways: by using the Quotient Rule and by simplifying first. Show that your answers are equivalent. Which method do you prefer?

23-42 Differentiate.

23. Vex) = (2x3

+

3)(x4 - 2x)

24. feu) ⁼(u-2

+

U-3)(U5 - 2u2)

[ill

F(y) =

(;2 - :4

^)(y

⁺

^5y3)

26. y ⁼

JX

^{(x -} 1) 3x - 1 27. g(x) =--

2x

+

1

28. f(t) ⁼--22t 4

+

t

x

+

1

30. Y=-3---

x +x-2 x3

29. y =---2

I-x

v

^{3 -}

2vJi;

31. y=---- v

t3

+

t 32. Y= t4 - 2

t -

Jt

34. get) = --1/-3 -

t

B

C

36. y = A

+- +-

X x2

37 Y=

• 1

+ v'7

38. y=---ex 1

+

ex

I1IJf(x) =-- x x

+-

e

x

ax

+

b 42. f(x) =--

ex

+

d

43. The general polynomial of degree n has the form P(x) =a"x"

+

a,,-Ix,,-I

+ ... +

a2x2

+

alx

+

ao

ffi

^44-46 Find f'(x). Compare the graphs off and f' and use them to explain why your answer is reasonable.

44. f(x) =x/(x2 - 1)

45. f(x) =3Xl5 - 5x3

+

3 1

46. f(x) =x

+ -

x

ffi

47. (a) Use a graphing calculator orcomputer to graph the func- tion f(x) =x4 - 3x3 - 6x2

+

7x

+

30 in the viewing rectangle [-3, 5Jby [-10,50].

(b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of

f'.

(See Example 1 in Section 3.2.)

(c) Calculate f'(x) and use this expression, with a graphing device, to graph

f'.

Compare with your sketch in part (b).

ffi

48. (a) Use a graphing calculator orcomputer to graph the func- tion g(x) =X2/(X2

+

1) in the viewing rectangle [-4, 4J by [ -1, 1.5].

(b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of g'.(See Example 1 in Section 3.2.)

(c) Calculate g'(x) and use this expression, with a graphing device, to graph g'. Compare with your sketch in part (b).

49-50 Find an equation of the tangent line to the curve at the given point.

49. y =--,2x (1, 1) x

+

1

51. (a) The curve y ⁼1/(1

+

x2) iscalled awitch of Maria Agnesi. Find an equation of the tangent line to this curve at the point

(-1, n

ffi

(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.

~ (a) The curve y = x/(l

+

x2) is called a serpentine.

Find an equation of the tangent line to this curve at the point (3, 0.3).

ffi

(b) Illustrate part (a) by graphing thecurve and thetangent line on the same screen.

53-56 Find equations of the tangent line and normal line to the curve at the given point.

54. Y⁼ (1

+

2X)2, (1,9) 56. y = --,

JX

(4,0.4)

x

+

1 3x

+

1

55. y =-2--' (1,2) x

+

1

x2

59. f(x) = I

+

2x

I 60. f(x) = 3 - x

[MJ

The equation ofmotion of aparticle is s = t3 - 3t,where s isin meters and tis inseconds. Find

(a) the velocity and acceleration as functions oft, (b) the acceleration after 2s,and

(c) the acceleration when the velocity isO.

62. The equation of motion ofaparticle is

s=2t3 - 7t2

+

4t

+

I, where sisinmeters and t isin seconds.

(a) Find the velocity and acceleration asfunctions of t.

(b) Find the acceleration after Is.

~ (c) Graph the position, velocity, and acceleration functions onthe same screen.

~ Suppose that f(5) = 1,1'(5) =6, g(5) = -3, and g'(5) =2.

Find the following values.

(a) (fg)'(5) (b) (f/g)'(5)

(c) (g/f)'(5)

64. Find h'(2), given thatf(2) = -3, g(2) =4,1'(2) = -2, and g'(2) =7.

(a) hex) =5f(x) - 4g(x)

(c) hex) =

~i:~

^{(b) hex)(d) hex) ==f(x)}I^g(x)

+

^g(x)f(x)

67. If

f

and 9are the functions whose graphs areshown, let u(x) =f(x)g(x) and vex) =f(x)/g(x).

(a) Find u'(l). (b) Find v'(5).

68. Let P(x) ⁼F(x)G(x) and Q(x) ⁼F(x)/G(x), where F and G are the functions whose graphs are shown.

(a) Find P'(2). (b) Find Q'(7).

y

\ ..-

\

^'\.

_/

^F_

-

./V

"-...

./

./ GI'---.

f-l '-...

0 1

69. If9isa differentiable function, find an expression for the derivative of each of the following functions.

x g(x)

(a) y =xg(x) (b) y =-(-) (c) y =-

9 x x

I7QJ

^If

f

is a differentiable function, find an expression for the derivative of each of the following functions.

(b) y = f(x) x2

x2

(c)y=-

f(x)

I

+

xf(x)

(d) y =

j;

ITIJ

Find the points on the curve y =2x3

+

3x2 - l2x

+

I where the tangent ishorizontal.

72. Forwhat values ofx does the graph of

f(x) ⁼x3

+

3x2

+

X

+

3have ahorizontal tangent?

73. Show that the curve y =6x3

+

5x - 3 has no tangent line with slope 4.

74. Find anequation of the tangent line to the curve y ⁼x ~ that isparallel to the liney =I

+

3x.

75. Find equations of both lines that are tangent to the curve y = I

+

x3 and areparallel to the line 12x - y = I.

76. Find equations of the tangent lines to the curve x-I

y=~

77. Find an equation of the normal line to the parabola y =x2 - 5x

+

4 that isparallel to the line x - 3y = 5.

78. Where does the normal line tothe parabola y ⁼x - x2 atthe point (I, 0)intersect the parabola asecond time? Illustrate with a sketch.

~ Draw adiagram to show that there are two tangent lines to the parabola y =x2 that pass through the point (0, -4). Find the coordinates of the points where these tangent lines inter- sect the parabola.

80. (a) Find equations of both lines through the point (2, - 3) that are tangent to the parabola y =x2

+

x.

(b) Show that there isno line through the point (2,7)that is tangent to the parabola. Then draw a diagram to seewhy.

lID:J

(a) Use the Product Rule twice to prove that iff, g,and hare

differentiable, then (fgh)' =f'gh

+

fg' h

+

f9h'.

(b) Taking

f

=9=h in part (a), show that

~ [J(X)J3 = 3[J(x)J21'(x)

(c) Use part (b) to differentiate y = (x4

+

3x3

+

l7x

+

82)3.

82. Find the nth derivative of each function bycalculating the first few derivatives and observing the pattern that occurs.

(a) f(x)

=

x" (b) f(x)

=

llx

83. Find a second-degree polynomial Psuch that P(2) =5, P'(2) =3,and P"(2) =2.

84. The equation y"

+

y' - 2y =x2 is called adifferential equation because it involves an unknown function y and its derivatives y' and y". Find constants A,B,and Csuch that the function y =Ax2

+

Bx

+

Csatisfies this equation. (Differen- tial equations will bestudied in detail inChapter 10.)

85. Find a cubic function y ⁼ax3

+

bx2

+

ex

+

d whose graph has horizontal tangents at the points (-2,6) and (2,0).

86. Find a parabola with equation y =ax2 + bx + ethat has slope 4at x = I, slope - 8 atx = -I, and passes through the point (2, 15).

87. In this exercise we estimate the rate at which the total personal income is rising in the Richmond-Petersburg, Virginia, metro- politan area. In 1999, the population of this area was 961,400, and thepopulation wasincreasing at roughly 9200 people per year.The average annual income was $30,593 per capita, and this average was increasing at about $1400 peryear (a little above the national average ofabout $1225 yearly). Use the Product Rule and these figures toestimate therate at which total personal income was rising inthe Richmond-Petersburg area in 1999. Explain the meaning of each term in the Product Rule.

88. Amanufacturer produces bolts ofa fabric with a fixedwidth.

The quantity qof this fabric (measured in meters) that issold is afunction of the selling pricep (in dollars per meter), sowe can write q =f(p). Then the total revenue earned with selling price pis R(p)

=

pf(p).

(a) What does it mean tosay that f(20) = 10,000 and 1'(20) = -350?

(b) Assuming the values in part (a), find R'(20) and interpret your answer.

{2 - x if x ~ 1

f(x) = x2 - 2x

+

2 if x

>

1

{-1-2X ifx<-I

g(x) = x2 if -1~x ~ I

x ifx>1

Give a formula forg' and sketch the graphs of 9 and g'.

91. (a) For what values ofx isthe functionf(x) ⁼1x2 - 91differ- entiable? Find a formula forf'.

(b) Sketch the graphs of

f

and

1'.

92. Where isthe function hex) =

I

x-II

+

1x

+

21 differenti- able? Give aformula for h' and sketch thegraphs ofhand h'.

~ For what values of aand b isthe line 2x

+

y ⁼btangent to the parabola y ⁼ax2 when x ⁼2?

94. (a) If F(x) =f(x) g(x), where f and 9have derivatives of all orders, show that F" ⁼f"g + 21'g' +fg".

(b) Find similar formulas for Fin and F(4).

(c) Guess a formula forFC").

95. Find the value of c such that theline y =~x

+

6is tangent to the curve y =

e£.

() {X2 if x ~ 2

f

x =

mx

+

b if x>2

Find the values of m and bthat make

f

differentiable everywhere.

97. An easy proof of theQuotient Rule can begiven ifwemake the prior assumption that F'(x) exists, where F ⁼

fig.

Write

f

=Fg; then differentiate using theProduct Rule and solve the resulting equation for F'.

98. Atangent line isdrawn tothe hyperbola xy ⁼c at a point P.

(a) Show that the midpoint of the line segment cut from this tangent line by the coordinate axes isP.

(b) Show that the triangle formed by the tangent line and the coordinate axes always has the same area, no matter where Pislocated on the hyperbola.

X1000 - 1

~ Evaluate lim ----.

x~1 x - I

100. Draw a diagram showing two perpendicular lines that intersect on the y-axis and are both tangent tothe parabola y =x2•

Where dothese lines intersect?

10I. Ife >

L

how many lines through the point (0,c) are normal lines to the parabola y =x2? What ife ~ ~?

102. Sketch the parabolas y ⁼x2 and y ⁼x2 - 2x

+

2.Do you think there is aline that istangent to both curves? Ifso,find its equation. If not, why not?

SOLUTION Here wedivide numerator and denominator by x:

1imcosx

.\"----1>0

x cos x cos x

Jimxcotx

=

tim -.--

=

Jim-. -

=

x•...•0 x•...•0 sInx x•...•O sInx l' sinx

IID--

x----').o X

G

^EXERCISES

I. f(x) =x - 3 sin x 3. y =sinx + 10 tan x 5. g(') =

,3

^cost

7. h(e) = ecsc e- cot e

[1;]y=

x

2 - tanx

2. f(x) =xsin x 4. y = 2csc x + 5 cos x 6. g(t) =4 sec , + tan' 8. y =u(a cos u + b cotu)

I + sinx 10. y =

x + cos x sec

e

II.f( fJ)=_I_+_s_ec_e I - sec x

12. y =

tanx 13. y =-2-sinx

X

17. Prove that -d (csc x) ⁼ -csc x cotx.

dx

18. Prove that -d (sec x) =sec x tanx.

dx

19. Prove that -d (cot x) = -csc2x.

dx

20. Prove, using the definition of derivative, that if f(x) =cos x,

then .f'(x) =-sin x.

21-24 Find an equation of the tangent line tothe curve at the given point.

24. y = . I , (0, 1)

smx + cos x

25. (a) Find an equation of the tangent line to the curve y =2x sin x at the point (71/2, 7T).

~ (b) Illustrate part (a) bygraphing thecurve and the tangent line on the same screen.

26. (a) Find anequation of the tangent line to thecurve y =sec x - 2 cosx at the point (7T/3, 1).

~ (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.

27. (a) If f(x) =sec x - x, findf'(x).

~ (b) Check to see that your answer topart (a) isreasonable by graphing both

f

and

f'

for

I

x

I <

7T/2.

28. (a) Iff(x) =~ sinx, findf'(x).

ffi

^{(b) Check to see that} your answer topart (a) isreasonable by graphing both f and f' for 0:S X :S 27T.

~ IfH(e) = esin e,find H'(e) and W(e).

30. If f(x) ⁼secx, find f"(7T/4).

() tanx - I fx=---

secx

(b) Simplify the expression for f(x) by writing it interms of sinxand cosx, and then find f'(x).

(c) Show that your answers toparts (a) and (b) are equivalent.

h(x) = cosx f(x)

Find (a) g'(7T/3) and (b) h'(7T/3).

InJ

^Forwhat values ofx does thegraph of f(x) =x + 2sinx have ahorizontal tangent?

34. Find the points on the curve y =(cos x)/(2 + sinx) atwhjch the tangent ishorizontal.

35. Amass ona spring vibrates horizontally ona smooth level surface (see the figure). Its equation ofmotion is x(t) = 8 sint,where t isin seconds andx in centimeters.

(a) Find the velocity and acceleration at time t.

(b) Find the position, velocity, and acceleration ofthe mass at time t=271/3. Inwhat direction isit moving atthat time?

equilibrium position

1

ffi

^{36. An} elastic band ishung onahook and amass ishung onthe lower end of the band. When the mass is pulled downward and then released, itvibrates vertically. The equation of motion is s =2 cost

+

3 sint,t ;. 0, where s ismeasured incentimeters and tinseconds. (Take the positive direction to be downward.)

(a) Find the velocity and acceleration attime t.

(b) Graph the velocity and acceleration functions.

(c) When does the masspass through the equilibrium position for the firsttime?

(d) How farfrom itsequilibrium position does themass travel?

(e) When is the speed the greatest?

37. Aladder 610 long rests against a vertical wall. Let 0be the angle between the top ofthe ladder and the wall and letx be the distance from the bottom ofthe ladder tothe wall. If the bottom ofthe ladder slides away from the wall, how fast does xchange with respect to 0 when 0 = 71/3?

38. An object with mass m isdragged along a horizontal plane by aforce acting along arope attached to the object. If the rope makes anangle 0 with the plane, then the magnitude of the force is

F= f.Lmg

f.Lsin 0

+

cos 0

where f.Lis aconstant called the coefficient oijriction.

(a) Find the rate of change of F with respect to O.

(b) When isthis rate of change equal to O?

(c) Ifm=20 kg,9 = 9.8m/s2, andf.L =0.6, draw thegraph of F asa function of 0 and use itto locate the value of 0 for which dF/dO =O.Isthe value consistent with your answer topart (b)?

sin 3x 39.1im--

x-a x

sin4x 40.lim--

x~o sin 6x

IiQ

lim tan 6t 42. lim cos0 - 1

HO sin 2t ^e~o sin0

43. lim sin(cos 0) sin23t

44. lim---

e~o sec0 ^(_0 t2

~ lim sinO 46. . sin(x2)

1110---

e~o 0

+

tan0 x_a x

47. lim I- tanx sin(x - I)

48. lim

x--"'7T/4 sinx - cosx ^x~1 x2

+

^{X -} 2

49. Differentiate each trigonometric identity to obtain a new (or familiar) identity.

sinx I

(a) tanx =-- (b) secx =--

cos x cos x

1

+

cot x (c) sinx

+

cos x =----

cscx

50. Asemicircle with diameter PQ sitson an isosceles triangle PQR toform aregion shaped like atwo-dimensional ice- cream cone, as shown inthe figure. IfA(O) isthe area of the semicircle and B(O) is the area of the triangle, find

lim A(O)

e~o+ B(O)

[Q

The figure shows a circular arc of length s and a chord of length d, both subtended by a central angle O.Find

lim !...-

e~o+ d

=================~

TH E CHAI N RULE

Suppose you are asked todifferentiate the function

F(x) =

R+1

The differentiation formulas you learned in the previous sections of this chapter do not enable you to calculate F'(x).

If we define £ to be 0 when Llx =0, then £ becomes acontinuous function of Llx. Thus, for a differentiable functionj, we can write

and £ is a continuous function of Llx. This property of differentiable functions is what enables us to prove the Chain Rule.

PROOF OF THE CHAIN RULE Suppose u

=

g(x) is differentiable at a and y

=

f(u) isdifferen- tiable atb =g(a). IfLlx is an increment inx and Llu and Lly are the corresponding incre- ments in u and y, then we can use Equation 5 towrite

where £2 ----0>0as Llu ----0>O.If we now substitute the expression forLlu from Equation 6 into Equation 7, we get

Lly = [f'(b)

+

^{£2] [g'(a)}

+

^{£J] Llx}

Lly

Llx =[f'(b)

+

£2][g'(a)

+ £J]

As Llx ----0>0, Equation 6shows that Llu ----0>O.So both £, ----0>0 and £2 ----0>0as Llx ----0>O.

Therefore

dy

=

lim ~y

=

lim [f'(b)

+

^£2][g'(a)

+ £1]

dx ilx-->O uX ilx-->O

§

^EXERCISES

1-6 Write the composite function in the form f(g(x». [Identify the

inner function u =g(x) and theouter function y =f(u).] Then find

the derivative dy/dx.

I.y ⁼sin4x

3. y =(l - X2)IO

[I]

Y ⁼Jsinx

2. y =J4

+

3x 4. y = tan(sinx) 6. y =sin

JX

15.Y=x sec kx 16. y ⁼3 cot(nfJ)

17. g(x) =(l

+

4X)5(3

+

x - X2)8 18. h(t) ⁼(t4 - 1)3(t3

+

1)4

~ Y=(2x - W(8x2 - 5)-3 20. y ⁼(x2

+

1).ijx2

+

2

( 2

+ 1)3

21. Y = _x__

x2 - 1

7.F(x) =(x3

+

4xr 9. F(x) = ~l

+

2x

+

x3

1

II. get) = (4 )l

t

+

1 13. Y= cos(a3

+

Xl)

8. F(x) =(x2 - X

+

1)3 10.f(x) =(I

+

X4)2/3

x 24. f(x) = ~3

'017 -.JX

~-I 25. F(z) = --

z

+

1

(y _ 1)4 26. G(y) =

(l +

2y)5 r

27.y= ~

yr2

+

1

COS7TX 28. Y⁼

sin TTX

+

cos 7TX

30. G(y) =

(~)5

_y+1

32. Y=tan2(38) I 34. Y =xsin-

x (1 - COS

2X)4

35. y= ----

I

+

cos 2x 36.f(t) =

-V '

^t

t-

+

4 38. Y=

(ax + ji2+jii)'2

39. Y⁼[x2

+

(I - 3xn3 40. y ⁼sin(sin(sin x))

41. y

= .jx + jX

42. Y

= Jx + .jx + JX

43. g(x)

=

^(2r sin rx

+

n)P 44. y

=

cos4(sin3x)

45. y ⁼cos.)sin(tan 7TX) 46. y ⁼[x

+

(x

+

sin2xn4

48. Y=sin2( 7Tt) 50. y= ~4x

yX

+

1

51-54 Find an equation ofthe tangent line to the curve atthe given point.

51. Y = ~,8 (4,2) 52. Y=sinx

+

cos 2x, (7T/6,1) y4

+

3x

53. y = sin(sin x), (7T, 0) 54. Y=.j5+X2, (2,3)

I

55. (a) Find anequation of the tangent line tothe curve y ⁼tan(^7Tx2/4) at thepoint (1, 1).

(b) Illustrate part (a) bygraphing the curve and the tangent line on the same screen.

56. (a) The curve y =

I

x

I/~

iscalled abullet-nose curve.

Find anequation ofthe tangent line to this curve atthe point (I, 1).

(b) Illustrate part (a)by graphing the curve and the tangent line onthe same screen.

57. (a) Iff(x) =x~, findf'(x).

~ (b) Check to seethat your answer topart (a) isreasonable by comparing the graphs of

f

and

f'.

~ 58. The function f(x) =sin(x

+

sin2x), 0 ~ x ~ 7T, arises in applications to frequency modulation (FM) synthesis.

(a) Use a graph of

f

produced by a graphing device tomake arough sketch ofthe graph of

f'.

(b) Calculate f'(x) and use this expression, with a graphing device, tograph f'. Compare with your sketch inpart (a).

m

^{Find all}points on the graph ofthe function

f(x) = 2 sin x

+

sin2x atwhich the tangent line ishorizontal.

60. Find thex-coordinates of all points onthe curve

y =sin2x - 2sin xatwhich the tangent line ishorizontal.

m

^{IfF(x) =f(g(x», where f( -2) = 8,1'( -2) =4,1'(5) = 3,}

g(5) =-2,and g'(5) =6, find F'(5).

62. If hex) = .)4

+

3f(x), where f(l) = 7and

I'

(l) = 4, find 17'(1).

x f(x) g(x) f'(x) g'(x)

I 3 2 4 6

2 I 8 5 7

3 7 2 7 9

(a) If hex) =f(g(x)), find 17'(1).

(b) If H(x) =g(f(x), find H'(l).

64. Let

f

and 9 bethe functions in Exercise 63.

(a) If F(x) =f(f(x», find F'(2).

(b) If G(x) =g(g(x»), find G'(3).

~ If

f

and 9are the functions whose graphs are shown, let u(x) =f(g(x», vex) = g(j(x», and w(x) =g(g(x». Find each derivative, ifit exists. Ifit does not exist, explain why.

(a) u'(1) (b) v'(l) (c) w'(l)

66. Iff isthe function whose graph isshown, let hex) ⁼f(f(x») and g(x) =f(x2). Use the graph off to estimate the value of each derivative.

(a) 17'(2) (b) g'(2)

~ Suppose f is differentiable on IR.Let F(x) =f(cos x) and G(x) =cos(f(x)). Find expressions for (a)F'(x) and (b)G'(x).

68. Suppose

f

isdifferentiable on IRand ais areal number.

Let F(x) =f(x^U) and G(x) =[f(x)Y Find expressions for(a)F'(x) and (b) G'(x).

69. Let rex) =f(g(h(x))), where h(l) = 2, g(2) = 3,h'(l) = 4, g'(2)

=

5,and1'(3)

=

6. Find r'(l).

70. If 9is atwice differentiable function andf(x) =xg(x2), find f" in terms of g, g', and g".

71. IfF(x) =f(3f(4f(x))), wheref(O) = 0 andl'(O) = 2, find F'(O).

72. IfF(x) =f(xf(xf(x))), wheref(l) = 2,f(2) = 3,f'(l) =4, 1'(2) =5,andl'(3) ⁼6,find F'(l).

73-74 Find the given derivative by finding the firstfew derivatives and observing thepattern that occurs.

~ The displacement of aparticle on a vibrating string is given by theequation

where s ismeasured incentimeters and tin seconds. Find the velocity of the particle after t seconds.

76. Ifthe equation of motion of a particle is given by s=Acos(wt

+

a),the particle is said toundergo simple harmonic motion.

(a) Find the velocity ofthe particle at time t.

(b) When is the velocity O?

77. ACepheid variable star isa star whose brightness alternately increases and decreases. The most easily visible such star is Delta Cephei, for which the interval between times of maxi- mum brightness is 5.4 days. The average brightness of this star is4.0 and its brightness changes by :!::0.35. In view of these data, the brightness of Delta Cephei at time t,where tismea- sured in days, has been modeled bythe function

B(t) =4.0

+

0.35 sin( 27Tt) 5.4

(a) Find the rate of change ofthe brightness after t days.

(b) Find, correct totwo decimal places, the rate of increase after one day.

78. In Example 4inSection 1.3 we arrived at amodel forthe length of daylight (in hours) inAnkara, Turkey, on the tth day of the year:

L(t) = 12

+

2.8 sin [ ;6~ (t - 80)]

Use this model to compare howthe number ofhours of day- light isincreasing inAnkara on March 21and May 21.

79. A particle moves along astraight line with displacement set), velocity vet), and acceleration aCt). Show that

aCt) =v(t)-dv ds

Explain the difference between the meanings of thederivatives dv/dt and dv/ds.

~ Air isbeing pumped into aspherical weather balloon. At any time t, the volume of theballoon is Vet) and its radius is r(t).

(a) What do thederivatives dV/dr anddV/dt represent?

(b) Express dV/dt interms of dr/dt.

[ill]

81. Computer algebra systems have commands that differentiate functions, but the form of the answer may not be convenient and sofurther commands may benecessary tosimplify the answer.

(a) Use aCAS to find the derivative inExample 5and com- pare with the answer inthat example. Then use the sim- plify command and compare again.

(b) Use a CAS to find thederivative inExample 6. What hap- pens if you use the simplify command? What happens if you use the factor command? Which form of the answer would be best for locating horizontal tangents?

and tosimplify the result.

(b) Where does the graph of

f

have horizontal tangents?

(c) Graph

f

and

I'

on the same screen. Are the graphs con- sistent with your answer topart (b)?

83. Use the Chain Rule toprove the following.

(a) The derivative of an even function is an odd function.

(b) The derivative ofan odd function isan even function.

84. Use the Chain Rule and the Product Rule togive an alternative proof of the Quotient Rule.

[Hint: Write f(x)/g(x) ⁼f(x)[g(xW'.]

-d (sin"x cos nx) =n sin"-Ix cos(n

+

Ox dx

(b) Find aformula for the derivative of y =cos"x cosnx that is similar tothe one inpart (a).

86. Suppose y =f(x) is a curve that always lies above the x-axis and never has ahorizontal tangent, where

f

isdifferentiable everywhere. For what value of y isthe rate of change of y5 with respect tox eighty times the rate of change ofy with respect tox?

~ Use the Chain Rule to show that if

e

^ismeasured indegrees, then

d 7T

- (sin

e) = --

cos

e

de 180

(This gives one reason for the convention that radian measure is always used when dealing with trigonometric functions in calculus: The differentiation formulas would notbe as simple if weused degree measure.)