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

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

51.How many tangent lines to the curve y ! x%! x " 1) pass through the point !1, 2"? At which points do these tangent lines touch the curve?

52.Find equations of the tangent lines to the curve

x ! 1 y ! x " 1

that are parallel to the line x ! 2y ! 2.

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

54.A manufacturer produces bolts of a fabric with a fixed width. The quantity q of this fabric (measured in yards) that is sold is a function of the selling price p (in dollars per yard), so we can

write q ! f ! p". Then the total revenue earned with selling price p is R! p" ! pf ! p".

(a)What does it mean to say that f !20" ! 10,000 and f %!20" ! !350?

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

55.(a) Use the Product Rule twice to prove that if f , t, and h are differentiable, then ! fth"% ! f %th " ft%h " fth%.

(b)Taking f ! t ! h in part (a), show that

dxd & f ! x"'3 ! 3& f ! x"'2 f %!x"

(c)Use part (b) to differentiate y ! e3x.

56.(a) If F! x" ! f ! x"t! x", where f and t have derivatives of all orders, show that F ' ! f 't " 2f %t% " ft'.

(b)Find similar formulas for F * and F !4".

(c)Guess a formula for F!n".

57.Find expressions for the first five derivatives of f ! x" ! x2ex. Do you see a pattern in these expressions? Guess a formula for f !n"! x" and prove it using mathematical induction.

58.(a) If t is differentiable, the Reciprocal Rule says that

Use the Quotient Rule to prove the Reciprocal Rule.

(b)Use the Reciprocal Rule to differentiate the function in Exercise 18.

(c)Use the Reciprocal Rule to verify that the Power Rule is valid for negative integers, that is,

dxd !x!n" ! !nx!n!1

for all positive integers n.

3.3

N A review of the trigonometric functions is given in Appendix D.

DERIVATIVES OF TRIGONOMETRIC FUNCTIONS

Before starting this section, you might need to review the trigonometric functions. In particular, it is important to remember that when we talk about the function f defined for all real numbers x by

f !x" ! sin x

TEC Visual 3.3 shows an animation of Figure 1.

FIGURE 1

cise 14 in Section 2.8), then it looks as if the graph of f % may be the same as the cosine curve (see Figure 1).

N We have used the addition formula for sine. See Appendix D.

Let’s try to confirm our guess that if f !x" ! sin x, then f %! x" ! cos x. From the definition of a derivative, we have

Two of these four limits are easy to evaluate. Since we regard x as a constant when computing a limit as h l 0, we have

2

lim

+l0

sin

+

+

! 1

FIGURE 2

N We multiply numerator and denominator by cos + " 1 in order to put the function in a form in which we can use the limits we know.

We know that lim +l0 1 ! 1 and lim +l0 cos + ! 1, so by the Squeeze Theorem, we have

But the function !sin +"%+ is an even function, so its right and left limits must be equal. Hence, we have

so we have proved Equation 2.

We can deduce the value of the remaining limit in (1) as follows:

192 |||| CHAPTER 3 DIFFERENTIATION RULES

3

lim

$

l0

cos

$ $

" 1 ! 0

N Figure 3 shows the graphs of the function of Example 1 and its derivative. Notice that y# ! 0 whenever y has a horizontal tangent.

5

y» y

_5

FIGURE 3

V EXAMPLE 1 Differentiate y ! x2 sin x.

SOLUTION Using the Product Rule and Formula 4, we have

Using the same methods as in the proof of Formula 4, one can prove (see Exercise 20) that

The tangent function can also be differentiated by using the definition of a derivative,

but it is easier to use the Quotient Rule together with Formulas 4 and 5: dxd !tan x" ! dxd #cossin xx $

cos x dxd !sin x" " sin x dxd !cos x" ! cos2x

cos x ! cos x " sin x !"sin x" ! cos2x

cos2x ! sin2x

!cos2x

!cos12x ! sec2x

N When you memorize this table, it is helpful to notice that the minus signs go with the derivatives of the “cofunctions,” that is, cosine, cosecant, and cotangent.

FIGURE 4

The horizontal tangents in Example 2

0

4

s

The derivatives of the remaining trigonometric functions, csc, sec, and cot, can also be found easily using the Quotient Rule (see Exercises 17–19). We collect all the differentiation formulas for trigonometric functions in the following table. Remember that they are valid only when x is measured in radians.

Trigonometric functions are often used in modeling real-world phenomena. In particular, vibrations, waves, elastic motions, and other quantities that vary in a periodic manner can be described using trigonometric functions. In the following example we discuss an instance of simple harmonic motion.

V EXAMPLE 3 An object at the end of a vertical spring is stretched 4 cm beyond its rest position and released at time t ! 0. (See Figure 5 and note that the downward direction is positive.) Its position at time t is

3.3EXERCISES

M

M

17.Prove that dxd !csc x" ! "csc x cot x.

18.Prove that dxd !sec x" ! sec x tan x.

19.Prove that dxd !cot x" ! "csc2x.

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 to the curve at the given point.

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

28.(a) If f !x" ! ex cos x, find f #!x" and f '!x".

;(b) Check to see that your answers to part (a) are reasonable by graphing f, f #, and f '.

29.If H!$" ! $ sin $, find H#!$" and H'!$".

30.If f !x" ! sec x, find f '!%%4".

196|||| CHAPTER 3 DIFFERENTIATION RULES

31.(a) Use the Quotient Rule to differentiate the function

f !x" ! tan x " 1 sec x

(b)Simplify the expression for f !x" by writing it in terms of sin x and cos x, and then find f #!x".

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

32.Suppose f !%%3" ! 4 and f #!%%3" ! "2, and let

Find (a) t#!%%3" and (b) h#!%%3".

33.For what values of x does the graph of f !x" ! x ! 2 sin x have a horizontal tangent?

34.Find the points on the curve y ! !cos x"%!2 ! sin x" at which the tangent is horizontal.

35.A mass on a spring vibrates horizontally on a smooth level surface (see the figure). Its equation of motion is

x!t" ! 8 sin t, where t is in seconds and x in centimeters.

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

(b) Find the position, velocity, and acceleration of the mass at time t ! 2%%3. In what direction is it moving at that time?

37. A ladder 10 ft long rests against a vertical wall. Let $ be the angle between the top of the ladder and the wall and let x be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does x change with respect to $ when $ ! %%3?

38.An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the

rope makes an angle $ with the plane, then the magnitude of the force is

)W

F ! ) sin $ ! cos $

where ) is a constant called the coefficient of friction.

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

(b) When is this rate of change equal to 0?

;(c) If W ! 50 lb and ) ! 0.6, draw the graph of F as a func-

;36. An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the mass is pulled downward and then released, it vibrates vertically. The equation of motion is s ! 2 cos t ! 3 sin t, t * 0, where s is measured in centimeters and t in seconds. (Take the positive direction to be downward.)

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

(b)Graph the velocity and acceleration functions.

(c)When does the mass pass through the equilibrium position for the first time?

(d)How far from its equilibrium position does the mass travel?

(e)When is the speed the greatest?

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

(a) tan x ! sin x cos x

(b) sec x ! cos1 x

(c) sin x ! cos x ! 1 ! cot x csc x

50.A semicircle with diameter PQ sits on an isosceles triangle PQR to form a region shaped like a two-dimensional ice-