Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
9
Transcribed Image Text:7:02
Exercises
In Exercises 1-20, find the derivative.
1. y = x lnx
3. y = 2¹
5. yIn(9x2-8)
7. y=(In x)2
9. y)2
11. yIn(In x)
13, y=(In(lnx))³
25. f(x)=6, x=2
27. s(t)=39, -2
29. f(x)=51²-2x, x=1
31. s(r) In (8-4), 1=1
33. R(2)=logs (2²+7). 2-3
43, y =
x(x+2)
V (2x + 1)(3x + 2)
44. y(x+1)(x4 + 2)(x³ +3)²
53. y
55. y
2, y rint-t
4. yIn(x³)
=
10. y=
cosh²(9-31)
coshx+1
coth!
1+ tanh t
59, y = sinh(lnx)
57. y =
6. y=In(15¹)
8. y=x² Inx
In Exercises 25-36, find an equation of the tangent line at the point
indicated.
61. y = tanh(e)
63. y = sech (√x)
65, y sech x coth x
12. y In(cotx)
14. y
35. f(w) = log₂ w, w
36. y log₂(1 +4x), x=4
In Exercises 37-44, find the derivative using logarithmic differentiation
as in Example 5.
37. y(x+5)(x +9)
39. y(x-1)(x-12)(x+7)
x(x²+1)
41. y=√√x+1
In x
=
X
In ((Inx)³)
In Exercises 45-50, find the derivative using either method of Ex-
ample 6
45. f(x)=x³x
46. f(x)=x-3¹
47. f(x)=
48. f(x)=x²
49. f(x)=x0.x
50. f(x)
In Exercises 51-74, calculate the derivative.
51. y
sinh(9x)
26. y=(√2), x=8
28. yx-2, x=1
30. s(1) Inr, 1=5
32. f(x)=In(x²), x=4
34. y=In(sin.x), x==
38. y (3x+5)(4x +9)
x(x + 1)³
40, y =
y = (3x-1)²
42. y = (2x + 1)(4x²)√√x-9
52. y
sinh(x2)
54. y = tinh(y2 +1)
56. y = sinh xanh x
58. y (In(cosh x))5
60, y
oth z
62. y sinh(cosh³ x)
64. y In(coth.x)
HÀNH V
66. y=x²
Further Insights and Challenges
84. (a) Show that if f and g are differentiable, then
182 CHAPTER 3 DIFFERENTIATION
f'(x) g'(x)
In(f(x)g(x)) = f(x)
g(x)
SECTION 3.9
4
dx
(b) Give a new proof of the Product Rule by observing that the left-
hand side of Eq. (4) is equal to f(x)g(x))
-
f(x)g(x)
15. yIn ((x + 1)(2x +9))
17. y = 11*
2-3-*
X
d
23. log (sint)
dr
19. y =
In Exercises 21-24, compute the derivative.
21. f'(x), f(x) = log₂x
67. y = cosh¹ (3x)
69. y = (sinh…!(2
77.
d
16. y = In (5+1)
18 y74x-x²
Derivatives of General Exponential and Logarithmic Functions
68. y = tanh−1(e* trong
70. y (esch-13x)4
72. y = sinh−'(v/24
74. y = In(tanh
71. yosh
73. y tanh(Inf)
In Exercises 75-77, prove the formula.
d
75.(coth.x) =
-(coth x)=-csch² x
dx
cosh"
20. y 16x
22. f'(3), f(x) = log x
d
24.log(+2¹)
fort > 1
√√2-1
78.
Use the formula (In f(x)) = f'(x)/f(x) to show than
and In(2x) have the same derivative. Is there a simpler explanatie
this result?
79. According to one simplified model, the purchasing power of a
lar in the year 2000+ is equal to P(r) = 0.68(1.04) (in 1983
lars). Calculate the predicted rate of decline in purchasing powe
cents per year) in the year 2020.
d
76. sinh…,=
dr
80. The energy E (in joules) radiated as seismic waves by an e
quake of Richter magnitude M satisfies log10 E = 4.8 +1.5M.
(a) Show that when M increases by 1, the energy increases by a f
of approximately 31.5.
(b) Calculate dE/dM.
85. Use the formula log x=
81. Show that for any constants M, k, and a, the function
k(t-a)
x) =ẩM (1 + ranh (Â(tz)))
y(t)
2
satisfies the logistic equation:=k(1-2).
82. Show that V(x) = 2 In(tanh(x/2)) satisfies the Pois
Boltzmann equation V"(x) = sinh(V(x)), which is used to des
electrostatic forces in certain molecules.
d
83. The Palermo Technical Impact Hazard Scale P is used to qua
the risk associated with the impact of an asteroid colliding with
earth:
PE08)
P= log10 0.037
1
√2+
where p, is the probability of impact, T is the number of years
impact, and E is the energy of impact (in megatons of TNT). The
is greater than a random event of similar magnitude if P >0.
(a) Calculated P/dT, assuming that p; = 2x 10-5 and E= 2m
tons.
doc-0g-68-docs.googleusercontent.com
(b) Use the derivative to estimate the change in P if 7 increases fr
to 9 years.
log x=
loga for a, b>0 to verify the for
loga b
(In b)x
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning