Solutions To Selected Problems Chapter A Chapter C

Chapter B

Exercises B.0

1.

$80x^{12}y^{17}$
2.

$\dfrac{a}{16b^{7}}$
3.

$\dfrac{x^{3}}{16y^{22}z^{35}}$
4.

$x^{2}y^{4}z^{5}\sqrt[4]{z}=x^{2}y^{4}z^{21/4}$
5.

$3x(x^{2}+9x+3)$
6.

$\frac{5(x-1)}{3x^{\frac{1}{3}}}$
7.

$\frac{-5x+4}{2x^{\frac{1}{2}}(x+4)^{2}}$
8.

$6x(3x^{2}+2)^{3}(x^{2}-5)^{2}(7x^{2}-18)$
9.
- 8
  
  44
  
  $x^{2}-6x+8$
  
  $x^{2}+2x-4$
10.
- $-\frac{1}{3}$
  
  undefined
  
  $\dfrac{1}{\sqrt{x-2}-5}$
  
  $\sqrt{\dfrac{1}{x-5}-2}$
11.
- Possible solution: $f(x)=\frac{5}{x}$ and $g(x)=x+4$
  
  Possible solution: $f(x)=\left\lvert x\right\rvert$ and $g(x)=4-x^{2}$
  
  Possible solution: $f(x)=\sqrt{x-5}$ and $g(x)=(x+2)^{2}$
12.
- Possible solution: $f(x)=\sqrt[3]{x}$ , $g(x)=x^{2}$ , and $h(x)=2x+1$
  
  Possible solution: $f(x)=2x+1$ , $g(x)=\sqrt[3]{x}$ , and $h(x)=x^{2}$

Exercises B.1

1.

T
2.

Answers will vary.
3.

Answers will vary.
4.

Answers will vary.
5.

(a) $f\,^{\prime}(x)=0$ , (b) $y=6$
6.

(a) $f\,^{\prime}(x)=2$ , (b) $y=2x$
7.

(a) $f\,^{\prime}(x)=-3$ , (b) $y=4-3x$
8.

(a) $g^{\prime}(x)=2x$ , (b) $y=-4x-4$
9.

(a) $h^{\prime}(x)=2-2x$ (b) $y=1$
10.

(a) $f\,^{\prime\prime}(x)=6x-1$ , (b) $y=-7x+1$
11.

(a) $g^{\prime}(x)=\frac{1}{2\sqrt{x+3}}$ , (b) $y=\frac{x}{4}+\frac{7}{4}$
12.

(a) $r\mkern 1.35mu^{\prime}(x)=\frac{-1}{x^{2}}$ , (b) $y=-\frac{x}{4}-1$
13.

(a) $h^{\prime}(x)=-\dfrac{3}{2x\sqrt{x}}$ , (b) $y=-\dfrac{3x}{16}+\dfrac{9}{4}$
14.

(a) $f\,^{\prime}(x)=\dfrac{-1}{(s-2)^{2}}$ , (b) $y=-x+4$
15.

$f(x)=\sqrt{x}$ , $c=16$ .
16.

$f(x)=x^{4}$ , $c=3$
17.

$f(x)=\dfrac{1}{x}$ , $c=2$
18.

$f(x)=\cos x$ , $c=-\pi$ .
19.

$y=8.1(x-3)+16$
20.

$y=.248x+1.006$
21.

$y=-0.099(x-9)+1$
22.

$y=7.77(x-2)+e^{2}$ , or $y=7.77(x-2)+7.39$
23.

$y=.49(x-2)+\ln 2$
24.

$y=-0.05x+1$
25.
- Approximations will vary; they should match (c) closely.
  
  $f\,^{\prime}(x)=2x$
  
  At $(-1,0)$ , slope is $-2$ . At $(0,-1)$ , slope is 0. At $(2,3)$ , slope is 4.
26.
- Approximations will vary; they should match (c) closely.
  
  $f\,^{\prime}(x)=-1/(x+1)^{2}$
  
  At $(0,1)$ , slope is $-1$ . At $(1,0.5)$ , slope is $-1/4$ .
27.
28.
29.
30.
31.
32.
33.
- Approximately on $(-2,0)$ and $(2,\infty)$ .
  
  Approximately on $(-\infty,-2)$ and $(0,2)$ .
  
  Approximately at $x=0,\ \pm 2$ .
  
  Approximately on $(-\infty,-1)$ and $(1,\infty)$ .
  
  Approximately on $(-1,1)$ .
  
  Approximately at $x=\pm 1$ .
34.
- Approximately on $(-1.5,1.5)$ .
  
  Approximately on $(-\infty,-1.5)\cup(1.5,\infty)$ .
  
  Approximately at $x=\pm 1.5$ .
  
  On $(-\infty,-1)\cup(0,1)$ .
  
  On $(-1,0)\cup(1,\infty)$ .
  
  At $x=\pm 1$ and $x=0$ .
35.
36.

Approximately $-24$ .
37.

Approximately $0.54$ .
38.
- $(-\infty,\infty)$
  
  $(-\infty,-1)\cup(-1,1)\cup(1,\infty)$
  
  $(-\infty,5]$
  
  $[-\sqrt{5},\sqrt{5}]$
39.
- 1
  
  3
  
  Does not exist
  
  $(-\infty,-3)\cup(3,\infty)$

Exercises B.2

1.

Velocity
2.

Answers will vary.
3.

Linear functions.
4.

12
5.

$-17$
6.

$102$
7.

$f(10.1)$ is likely most accurate, as accuracy is lost the farther from $x=10$ we go.
8.

$-4$
9.

$6$
10.

decibels per person
11.

ft/s²
12.

ft/h
13.
- thousands of dollars per car
  
  It is likely that $P(0)<0$ . That is, negative profit for not producing any cars.
14.
- degrees Fahrenheit per hour
  
  It is likely that $T\mkern 1.35mu^{\prime}(8)>0$ since at 8 in the morning, the temperature is likely rising.
  
  It is very likely that $T(8)>0$ , as at 8 in the morning on July 4, we would expect the temperature to be well above 0.
15.

$f(x)=g^{\prime}(x)$
16.

$g(x)=f\,^{\prime}(x)$
17.

$g(x)=f\,^{\prime}(x)$
18.

$g(x)=f\,^{\prime}(x)$
19.

$f(6)=1$ , $f\,^{\prime}(6)=-\frac{3}{4}$
20.

Answers vary. Possible solution
21.

Answers vary. Possible solution
22.

$x=\pm 1$
23.

$f\,^{\prime}(x)=10x$
24.

$f\,^{\prime}(x)=3x^{2}-12x+12$
25.

$f\,^{\prime}(\pi)\approx 0$ .
26.

$f\,^{\prime}(9)\approx 0.1667$ .

Exercises B.3

1.

Power Rule.
2.

$1/x$
3.

One answer is $f(x)=10e^{x}$ .
4.

One answer is $f(x)=10$ .
5.

$f(x)$ , $g(x)$ , $h(x)$ , and $m(x)$
6.

Answers will vary.
7.

One possible answer is $f(x)=17x-205$ .
8.

Answers will vary.
9.

$f\,^{\prime}(x)$ is a velocity function, and $f\,^{\prime\prime}(x)$ is acceleration.
10.

lbs/ft².
11.

$f\,^{\prime}(x)=14x-5$
12.

$g^{\prime}(x)=42x^{2}+14x+11$
13.

$m^{\prime}(t)=45t^{4}-\frac{3}{8}t^{2}+3$
14.

$f\,^{\prime}(\theta)=9\cos\theta-10\sin\theta$
15.

$f\,^{\prime}(r)=6e^{r}$
16.

$g^{\prime}(t)=40t^{3}+\sin t+7\cos t$
17.

$f\,^{\prime}(x)=\frac{2}{x}-1$
18.

$p^{\prime}(s)=s^{3}+s^{2}+s+1$
19.

$h^{\prime}(t)=e^{t}-\cos t+\sin t$
20.

$f\,^{\prime}(x)=\frac{2}{x}$
21.

$f\,^{\prime}(t)=0$
22.

$g^{\prime}(t)=18t+6$
23.

$g^{\prime}(x)=24x^{2}-120x+150$
24.

$f\,^{\prime}(x)=-3x^{2}+6x-3$
25.

$f\,^{\prime}(x)=18x-12$
26.

$h^{\prime}(x)=3x^{2}-2$
27.

$f\,^{\prime}(x)=\dfrac{3}{2}\sqrt{x}-\dfrac{1}{2x\sqrt{x}}$
28.

$g^{\prime}(\theta)=-\sin\theta$
29.
30.

$\displaystyle\frac{\operatorname{d}\!}{\operatorname{d}\!x}(c)=\lim_{h\to 0}% \frac{c-c}{h}=\lim_{h\to 0}0=0$
31.

$a$ is $f$ , $b$ is $f\,^{\prime}$ , $c$ is $f\,^{\prime\prime}$
32.

$d$ is $f$ , $c$ is $f\,^{\prime}$ , $b$ is $f\,^{\prime\prime}$ , and $a$ is $f\,^{\prime\prime\prime}$
33.

$f\,^{\prime}(x)=6x^{5}$ $f\,^{\prime\prime}(x)=30x^{4}$ $f\,^{\prime\prime\prime}(x)=120x^{3}$ $f^{(4)}(x)=360x^{2}$
34.

$g^{\prime}(x)=-2\sin x$ $g^{\prime\prime}(x)=-2\cos x$ $g^{\prime\prime\prime}(x)=2\sin x$ $g^{(4)}(x)=2\cos x$
35.

$h^{\prime}(t)=2t-e^{t}$ $h^{\prime\prime}(t)=2-e^{t}$ $h^{\prime\prime\prime}(t)=-e^{t}$ $h^{(4)}(t)=-e^{t}$
36.

$p^{\prime}(\theta)=4\theta^{3}-3\theta^{2}$ $p^{\prime\prime}(\theta)=12\theta^{2}-6\theta$ $p^{\prime\prime\prime}(\theta)=24\theta-6$ $p^{(4)}(\theta)=24$
37.

$f\,^{\prime}(\theta)=\cos\theta+\sin\theta$ $f\,^{\prime\prime}(\theta)=-\sin\theta+\cos\theta$ $f\,^{\prime\prime\prime}(\theta)=-\cos\theta-\sin\theta$ $f^{(4)}(\theta)=\sin\theta-\cos\theta$
38.

$f\,^{\prime}(x)=f\,^{\prime\prime}(x)=f\,^{\prime\prime\prime}(x)=f^{(4)}(x)=0$
39.
- $v(t)=4t^{3}-8t$ , $a(t)=12t^{2}-8$
  
  $a(1.5)=19\ \text{ft/s}^{2}$
  
  $t=0$ sec and $t=\sqrt{2}$ sec
40.
- $v(t)=5e^{x}-5$ , $a(t)=5e^{x}$
  
  $a(2)=5e^{2}\ \text{ft/s}^{2}$
  
  $v(t)=0$ at $t=0$ sec, $a(0)=5\ \text{in/s}^{2}$
41.

Tangent line: $y=2(x-1)$
42.

Tangent line: $y=t+4$
43.

Tangent line: $y=x-1$
44.

Tangent line: $y=4$
45.

Tangent line: $y=\sqrt{2}(x-\frac{\pi}{4})-\sqrt{2}$
46.

Tangent line: $y=2x+3$
47.

$n=-3,2$
48.

The tangent line to $f(x)=e^{x}$ at $x=0$ is $y=x+1$ ; thus $e^{0.1}\approx y(0.1)=1.1$ .
49.

The tangent line to $f(x)=x^{4}$ at $x=3$ is $y=108(x-3)+81$ ; thus $(3.01)^{4}\approx y(3.01)=108(.01)+81=82.08$ .

Exercises B.4

1.

F
2.

F
3.

T
4.

Quotient Rule
5.

F
6.

Answers will vary.
7.

$\displaystyle\frac{\operatorname{d}\!}{\operatorname{d}\!x}(\cot x)$ $\displaystyle=\frac{\operatorname{d}\!}{\operatorname{d}\!x}\left(\frac{\cos x% }{\sin x}\right)$

$\displaystyle=\frac{\sin x(-\sin x)-(\cos x)(\cos x)}{(\sin x)^{2}}$

$\displaystyle=\frac{-[(\sin x)^{2}+(\cos x)^{2}]}{(\sin x)^{2}}$

$\displaystyle=\frac{-1}{(\sin x)^{2}}=-\csc^{2}x$
8.

$\displaystyle\frac{\operatorname{d}\!}{\operatorname{d}\!x}(\csc x)$ $\displaystyle=\frac{\operatorname{d}\!}{\operatorname{d}\!x}\left(\frac{1}{% \sin x}\right)$

$\displaystyle=\frac{\sin x\cdot 0-1\cdot(\cos x)}{(\sin x)^{2}}$

$\displaystyle=\frac{-\cos x}{(\sin x)^{2}}=-\csc x\cot x$
9.
- $f\,^{\prime}(x)=(x^{2}+3x)+x(2x+3)$
  
  $f\,^{\prime}(x)=3x^{2}+6x$
  
  They are equal.
10.
- $g^{\prime}(x)=4x(5x^{3})+2x^{2}(15x^{2})$
  
  $g^{\prime}(x)=50x^{4}$
  
  They are equal.
11.
- $h^{\prime}(s)=2(s+4)+(2s-1)(1)$
  
  $h^{\prime}(s)=4s+7$
  
  They are equal.
12.
- $f\,^{\prime}(x)=2x(3-x^{3})+(x^{2}+5)(-3x^{2})$
  
  $f\,^{\prime}(x)=-5x^{4}-15x^{2}+6x$
  
  They are equal.
13.
- $f\,^{\prime}(x)=\frac{x(2x)-(x^{2}+3)1}{x^{2}}$
  
  $f\,^{\prime}(x)=1-\frac{3}{x^{2}}$
  
  They are equal.
14.
- $g^{\prime}(x)=\frac{2x^{2}(3x^{2}-4x)-(x^{3}-2x^{2})(4x)}{4x^{4}}$
  
  $g^{\prime}(x)=1/2$
  
  They are equal.
15.
- $h^{\prime}(s)=\frac{4s^{3}(0)-3(12s^{2})}{16s^{6}}$
  
  $h^{\prime}(s)=-9/4s^{-4}$
  
  They are equal.
16.
- $f\,^{\prime}(t)=\frac{(t+1)(2t)-(t^{2}-1)(1)}{(t+1)^{2}}$
  
  $f(t)=t-1$ when $t\neq-1$ , so $f\,^{\prime}(t)=1$ .
  
  They are equal.
17.

$f\,^{\prime}(x)=\sin x+x\cos x$
18.

$f\,^{\prime}(t)=\frac{-2}{t^{3}}(\csc t-4)+\frac{1}{t^{2}}(-\csc t\cot t)$
19.

$H^{\prime}(y)=(y^{5}-2y^{3})(14y+1)+(5y^{4}-6y^{2})(7y^{2}+y-8)$
20.

$F\mkern 1.35mu^{\prime}(y)=\dfrac{8}{3}y^{5/3}+15y^{2/3}=\frac{\sqrt[3]{y^{2}}% (8y+45)}{3}$
21.

$g^{\prime}(x)=\frac{-12}{(x-5)^{2}}$
22.

$y^{\prime}=\dfrac{4-x}{2\sqrt{x}(x+4)^{2}}$
23.

$g^{\prime}(x)=\dfrac{\sqrt{x}+8}{2(\sqrt{x}+4)^{2}}$
24.

$g^{\prime}(t)=\frac{(\cos t-2t^{2})(5t^{4})-(t^{5})(-\sin t-4t)}{(\cos t-2t^{2% })^{2}}$
25.

$h^{\prime}(x)=-\csc^{2}x-e^{x}$
26.

$h^{\prime}(t)=14t+6$
27.

$f\,^{\prime}(x)=\frac{(x+2)(4x^{3}+6x^{2})-(x^{4}+2x^{3})(1)}{(x+2)^{2}}$
28.

$f\,^{\prime}(x)=-\dfrac{1}{x^{2}}+\dfrac{5}{2x^{3}\sqrt{x}}=\dfrac{-2x\sqrt{x}% +5}{2x^{3}\sqrt{x}}$
29.

$y^{\prime}=-2x-5-\dfrac{10}{x^{2}}=-\dfrac{2x^{3}+5x^{2}+10}{x^{2}}$
30.

$g^{\prime}(x)=-\frac{1+2x+3x^{2}}{(1+x+x^{2}+x^{3})^{2}}$
31.

$p^{\prime}(x)=-\dfrac{1}{x^{2}}-\dfrac{2}{x^{3}}-\dfrac{3}{x^{4}}=-\dfrac{x^{2% }+2x+3}{x^{4}}$
32.

$f\,^{\prime}(x)=7$
33.

$f\,^{\prime}(t)=5t^{4}(\sec t+e^{t})+t^{5}(\sec t\tan t+e^{t})$
34.

$f\,^{\prime}(x)=\frac{\sin^{2}(x)+\cos^{2}(x)+3\cos(x)}{(\cos(x)+3)^{2}}$
35.

$g^{\prime}(x)=0$
36.

$g^{\prime}(t)=12t^{2}e^{t}+4t^{3}e^{t}-\cos^{2}t+\sin^{2}t$
37.

$f\,^{\prime}(y)=y(2y^{3}-5y-1)(12y)+y(6y^{2}-5)(6y^{2}+7)+1(2y^{3}-5y-1)(6y^{2% }+7)=72y^{5}-64y^{3}-18y^{2}-70y-7$
38.

$F\mkern 1.35mu^{\prime}(x)=(8x-1)(x^{2}+4x+7)(3x^{2})+(8x-1)(2x+4)(x^{3}-5)+(8% )(x^{2}+4x+7)(x^{3}-5)$
39.

$h^{\prime}(x)=\frac{(t^{2}\cos t+2)(2t\sin t+t^{2}\cos t)-(t^{2}\sin t+3)(2t% \cos t-t^{2}\sin t)}{(t^{2}\cos t+2)^{2}}$
40.

$f\,^{\prime}(x)=2xe^{x}\tan x=x^{2}e^{x}\tan x+x^{2}e^{x}\sec^{2}x$
41.

$g^{\prime}(x)=2\sin x\sec x+2x\cos x\sec x+2x\sin x\sec x\tan x=2\tan x+2x+2x% \tan^{2}x=2\tan x+2x\sec^{2}x$
42.

$f\,^{\prime}(x)=1+\ln x$
43.

$y=2x+2$
44.

$y=-(x-\frac{3\pi}{2})-\frac{3\pi}{2}=-x$
45.

$y=4$
46.

$y=-9x+1$
47.

$x=3/2$
48.

$x=0$
49.

$f\,^{\prime}(x)$ is never 0.
50.

$x=-2,0$
51.

$f\,^{\prime\prime}(x)=2\cos x-x\sin x$
52.

$f^{(4)}(x)=-4\cos x+x\sin x$
53.

$f\,^{\prime\prime}(x)=\cot^{2}x\csc x+\csc^{3}x$
54.

$f^{(8)}=0$
55.

$1$
56.

$-3$
57.

$-4$
58.

$11$
59.

$-\dfrac{1}{25}$
60.

$\dfrac{1}{4}$
61.
- () $-\frac{7}{2}$ () $\frac{1}{18}$ () $-\frac{9}{2}$ () $\frac{15}{2}$

Exercises B.5

1.

T
2.

F
3.

F
4.

F
5.

T
6.

T
7.

$f\,^{\prime}(x)=10(4x^{3}-x)^{9}\cdot(12x^{2}-1)=(120x^{2}-10)(4x^{3}-x)^{9}$
8.

$f\,^{\prime}(t)=15(3t-2)^{4}$
9.

$g^{\prime}(\theta)=3(\sin\theta+\cos\theta)^{2}(\cos\theta-\sin\theta)$
10.

$h^{\prime}(t)=(6t+1)e^{3t^{2}+t-1}$
11.

$f\,^{\prime}(x)=4\big(x+\frac{1}{x}\big)^{3}\big(1-\frac{1}{x^{2}}\big)$
12.

$p^{\prime}(x)=12\left(x^{2}-\dfrac{1}{x^{2}}\right)^{5}\left(x+\dfrac{1}{x^{3}% }\right)$
13.

$f\,^{\prime}(x)=-3\sin(3x)$
14.

$g^{\prime}(x)=5\sec^{2}(5x)$
15.

$h^{\prime}(x)=(2\theta+4)\sec^{2}(\theta^{2}+4\theta)$
16.

$g^{\prime}(t)=(5t^{4}-\frac{1}{t^{2}})\cos(t^{5}+\frac{1}{t})$
17.

$h^{\prime}(t)=8\sin^{3}(2t)\cos(2t)$
18.

$p^{\prime}(t)=-3\cos^{2}(t^{2}+3t+1)\sin(t^{2}+3t+1)(2t+3)$
19.

$g^{\prime}(x)=2(\tan x\sec^{2}x-x\sec^{2}(x^{2}))$
20.

$w^{\prime}(x)=3x^{2}e^{x^{3}}(\sec e^{x^{3}})(\tan e^{x^{3}})$
21.

$f\,^{\prime}(x)=-\tan x$
22.

$f\,^{\prime}(x)=2/x$
23.

$f\,^{\prime}(x)=2/x$
24.

$g^{\prime}(t)=0$
25.

$r^{\prime}(x)=\dfrac{-6(x-1)}{x^{3}\sqrt{4x-3}}$
26.

$f\,^{\prime}(x)=\dfrac{12x(2x^{3}-1)(3x^{2}-5)^{3}(x^{2}+5x-2)}{(2x^{3}-1)^{4}}$
27.

$h^{\prime}(x)=200(2x+1)^{9}[(2x+1)^{10}+1]^{9}$
28.

$f\,^{\prime}(t)=\dfrac{-t^{4}}{(2t+1)(t+1)}$
29.

$F\mkern 1.35mu^{\prime}(x)=2(2x+1)(2x+3)^{2}(24x^{2}+26x+3)$
30.

$f\,^{\prime}(x)=5x^{2}\cos(5x)+2x\sin(5x)$
31.

$f\,^{\prime}(x)=5(x^{2}+x)^{4}(2x+1)(3x^{4}+2x)^{3}+(x^{2}+x)^{5}3(3x^{4}+2x)^% {2}(12x^{3}+2)$
32.

$g^{\prime}(t)=5\cos(t^{2}+3t)\cos(5t-7)-(2t+3)\sin(t^{2}+3t)\sin(5t-7)$
33.

$g^{\prime}(t)=10t\cos(\frac{1}{t})e^{5t^{2}}+\frac{1}{t^{2}}\sin(\frac{1}{t})e% ^{5t^{2}}$
34.

$f\,^{\prime}(x)=\dfrac{(5x-9)^{3}4\cos(4x+1)-\sin(4x+1)15(5x-9)^{2}}{(5x-9)^{6}}$
35.

$f\,^{\prime}(x)=\dfrac{\tan(5x)8(4x+1)-(4x+1)^{2}5\sec^{2}(5x)}{\tan^{2}(5x)}$
36.

$a^{\prime}(t)=7t^{2}e^{\tan(t^{2})}(2t^{2}\sec^{2}(t^{2})+3)$
37.

$y^{\prime}=\dfrac{-\cos x\sin x\cos(\cos^{2}x)}{\sqrt{\sin(\cos^{2}x)}}$
38.

$k^{\prime}(x)=-\sin(x\sin x^{3})(3x^{3}\cos x^{3}+\sin x^{3})$
39.

$f\,^{\prime}(x)=\frac{1}{2}x^{-1/2}-\frac{1}{2}x^{-3/2}=\frac{1}{2\sqrt{x}}-% \frac{1}{2\sqrt{x^{3}}}$
40.

$f\,^{\prime}(x)=\frac{1}{3}x^{-2/3}+\frac{2}{3}x^{-1/3}=\frac{1}{3\sqrt[3]{x^{% 2}}}+\frac{2}{3\sqrt[3]{x}}$
41.

$f\,^{\prime}(t)=\frac{-t}{\sqrt{1-t^{2}}}$
42.

$g^{\prime}(t)=\sqrt{t}\cos t+\frac{\sin t}{2\sqrt{t}}$
43.

$h^{\prime}(x)=1.5x^{0.5}=1.5\sqrt{x}$
44.

$f\,^{\prime}(x)=\pi x^{\pi-1}+1.9x^{0.9}$
45.

$g^{\prime}(x)=\frac{\sqrt{x}(1)-(x+7)(1/2x^{-1/2})}{x}=\frac{1}{2\sqrt{x}}-% \frac{7}{2\sqrt{x^{3}}}$
46.

$f\,^{\prime}(t)=\frac{1}{5}x^{-4/5}(\sec t+e^{t})+\sqrt[5]{t}(\sec t\tan t+e^{% t})$
47.

$15$
48.

$90$
49.
- () $6$ () $1$ () $-3$ () $1.5$
50.
- () $12$ () $2.5$ () $9$ () $35$
51.

$y=0$
52.

$y=15(t-1)+1$
53.

$y=-3(\theta-\pi/2)+1$
54.

$y=-5e(t+1)+e$
55.

In both cases the derivative is the same: $1/x$ .
56.

In both cases the derivative is the same: $k/x$ .
57.
Let $g(x)=-x$ . Then
- $f\circ g=f$ , so $f\,^{\prime}(-x)=f\,^{\prime}\circ g(x)=-f\,^{\prime}\circ g(x)g^{\prime}(x)=-% (f\circ g)^{\prime}(x)=-f\,^{\prime}(x)$
  
  $f\circ g=-f$ , so $f\,^{\prime}(-x)=f\,^{\prime}\circ g(x)=-f\,^{\prime}\circ g(x)g^{\prime}(x)=-% (f\circ g)^{\prime}(x)=f\,^{\prime}(x)$
58.

Let $h(x)=x^{-1}$ . Then $\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}\dfrac{f(x)}{g(x)}=\dfrac{% \operatorname{d}\!}{\operatorname{d}\!x}\left[f(x)\cdot h(g(x))\right]=\dfrac{% \operatorname{d}\!}{\operatorname{d}\!x}\left[f(x)\right]\cdot h(g(x))+f(x)% \cdot\dfrac{\operatorname{d}\!}{\operatorname{d}\!x}\left[h(g(x))\right]=f\,^{% \prime}(x)\cdot h(g(x))+f(x)\cdot h^{\prime}(g(x))\cdot g^{\prime}(x)=f\,^{% \prime}(x)[g(x)]^{-1}-f(x)[g(x)]^{-2}g^{\prime}(x)=\dfrac{f\,^{\prime}(x)g(x)-% f(x)g^{\prime}(x)}{[g(x)]^{2}}$
59.

$[f(g(x))]^{\prime\prime}=[f\,^{\prime}(g(x))g^{\prime}(x)]^{\prime}=[f\,^{% \prime}(g(x))]^{\prime}g^{\prime}(x)+f\,^{\prime}(g(x))g^{\prime\prime}(x)=f\,% ^{\prime\prime}(g(x))g^{\prime}(x)g^{\prime}(x)+f\,^{\prime}(g(x))g^{\prime% \prime}(x)=f\,^{\prime\prime}(g(x))[g^{\prime}(x)]^{2}+f\,^{\prime}(g(x))g^{% \prime\prime}(x)$
60.
- ^∘F/mph
  
  The sign would be negative; when the wind is blowing at 10 mph, any increase in wind speed will make it feel colder, i.e., a lower number on the Fahrenheit scale.
61.

$2xe^{x}\cot x+x^{2}e^{x}\cot x-x^{2}e^{x}\csc^{2}x$

Exercises B.6

1.

Answers will vary.
2.

The Chain Rule.
3.

T
4.

T
5.

$\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=\frac{-4x^{3}}{2y+1}$
6.

$\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=-\frac{y^{3/5}}{x^{3/5}}$
7.

$\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=\sin x\sec y$
8.

$\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=\frac{y}{x}$
9.

$\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=\frac{y}{x}$
10.

$-e^{x}x(x+2)e^{-y}$
11.

$-\frac{2\sin(y)\cos(y)}{x}$
12.

$-\frac{x}{y^{2}}$
13.

$\frac{1}{2y+2}$
14.

$\frac{x^{2}+2xy^{2}-y}{2x^{2}y-x+y^{2}}$
15.

$\frac{-\cos(x)(x+\cos(y))+\sin(x)+y}{\sin(y)(\sin(x)+y)+x+\cos(y)}$
16.

$-\frac{x}{y}$
17.

$-\frac{2x+y}{2y+x}$
18.

$\dfrac{e^{x}(x+1)}{e^{y}(y+1)}$
19.

$\dfrac{3x^{2}y\cos(x^{3})-\sin(y^{3})}{3xy^{2}\cos(y^{3})-\sin(x^{3})}$
20.

$\dfrac{y-4xy\sqrt{xy}}{2x^{2}\sqrt{xy}-x}$
21.

$\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=\frac{y(y-2x)}{x(x-2y)}$
22.

$\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=-\frac{y+2x}{2y+x}$
23.
- $y=0$
  
  $y=-1.859(x-0.1)+0.281$
24.
- $x=1$
  
  $y=-\frac{3\sqrt{3}}{8}(x-\sqrt{.6})+\sqrt{.8}\approx-0.65(x-0.775)+0.894$
  
  $y=1$
25.
- $y=4$
  
  $y=0.93(x-2)-\sqrt[4]{108}$
26.
- $y=-1/3x+1$
  
  $y=3\sqrt{3}/4$
27.
- $y=-\frac{1}{\sqrt{3}}(x-\frac{7}{2})+\frac{6+3\sqrt{3}}{2}$
  
  $y=\sqrt{3}(x-\frac{4+3\sqrt{3}}{2})+\frac{3}{2}$
28.
- $y=-\dfrac{3x}{4}-\dfrac{3}{2}$
  
  $y=\dfrac{7}{2}x-\dfrac{3}{2}$
29.

$\frac{\operatorname{d}\!^{2}y}{\operatorname{d}\!x^{2}}=\frac{(2y+1)(-12x^{2})% +4x^{3}\left(2\frac{-4x^{3}}{2y+1}\right)}{(2y+1)^{2}}$
30.

$\frac{\operatorname{d}\!^{2}y}{\operatorname{d}\!x^{2}}=\frac{3}{5}\frac{y^{3/% 5}}{x^{8/5}}+\frac{3}{5}\frac{1}{yx^{6/5}}$
31.

$\frac{\operatorname{d}\!^{2}y}{\operatorname{d}\!x^{2}}=\frac{\cos x\cos y+% \sin^{2}x\tan y}{\cos^{2}y}$
32.

$\frac{\operatorname{d}\!^{2}y}{\operatorname{d}\!x^{2}}=0$
33.

In each, $\frac{\operatorname{d}\!y}{\operatorname{d}\!x}=-\frac{y}{x}$ .

	$\displaystyle\frac{\operatorname{d}\!}{\operatorname{d}\!x}(\cot x)$	$\displaystyle=\frac{\operatorname{d}\!}{\operatorname{d}\!x}\left(\frac{\cos x% }{\sin x}\right)$
		$\displaystyle=\frac{\sin x(-\sin x)-(\cos x)(\cos x)}{(\sin x)^{2}}$
		$\displaystyle=\frac{-[(\sin x)^{2}+(\cos x)^{2}]}{(\sin x)^{2}}$
		$\displaystyle=\frac{-1}{(\sin x)^{2}}=-\csc^{2}x$

	$\displaystyle\frac{\operatorname{d}\!}{\operatorname{d}\!x}(\csc x)$	$\displaystyle=\frac{\operatorname{d}\!}{\operatorname{d}\!x}\left(\frac{1}{% \sin x}\right)$
		$\displaystyle=\frac{\sin x\cdot 0-1\cdot(\cos x)}{(\sin x)^{2}}$
		$\displaystyle=\frac{-\cos x}{(\sin x)^{2}}=-\csc x\cot x$