James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 3.6 Derivatives of Logarithmic Functions 223
3.6
Exercises
1. Explain why the natural logarithmic function y − ln x is used
much more frequently in calculus than the other logarithmic
functions y − log b x.
2–22 Differentiate the function.
2. f sxd − x ln x 2 x
3. f sxd − sinsln xd 4. f sxd − lnssin 2 xd
5. f sxd − ln 1 x
7. f sxd − log 10s1 1 cos xd
6. y − 1
ln x
8. f sxd − log 10sx
9. tsxd − lnsxe 22x d 10. tstd − s1 1 ln t
11. Fstd − sln td 2 sin t 12. hsxd − lnsx 1 sx 2 2 1d
13. Gsyd − ln
s2y 1 1d5
sy 2 1 1
14. Psvd − ln v
1 2 v
15. Fssd − ln ln s 16. y − ln | 1 1 t 2 t 3 |
17. Tszd − 2 z log 2 z
18. y − lnscsc x 2 cot xd
19. y − lnse 2x 1 xe 2x d 20. Hszd − lnÎ a 2 2 z 2
a 2 1 z 2
21. y − tanflnsax 1 bdg 22. y − log 2 sx log 5 xd
23–26 Find y9 and y99.
23. y − sx ln x 24. y − ln x
1 1 ln x
25. y − ln | sec x |
26. y − lns1 1 ln xd
27–30 Differentiate f and find the domain of f .
x
27. f sxd −
28. f sxd − s2 1 ln x
1 2 lnsx 2 1d
29. f sxd − lnsx 2 2 2xd 30. f sxd − ln ln ln x
31. If f sxd − lnsx 1 ln xd, find f 9s1d.
32. If f sxd − cossln x 2 d, find f 9s1d.
;
;
33–34 Find an equation of the tangent line to the curve at the
given point.
33. y − lnsx 2 2 3x 1 1d, s3, 0d
34. y − x 2 ln x, s1, 0d
35. If f sxd − sin x 1 ln x, find f 9sxd. Check that your answer is
reasonable by comparing the graphs of f and f 9.
36. Find equations of the tangent lines to the curve y − sln xdyx
at the points s1, 0d and se, 1yed. Illustrate by graphing the
curve and its tangent lines.
37. Let f sxd − cx 1 lnscos xd. For what value of c is
f 9sy4d − 6?
38. Let f sxd − log bs3x 2 2 2d. For what value of b is f 9s1d − 3?
39–50 Use logarithmic differentiation to find the derivative of the
function.
39. y − sx 2 1 2d 2 sx 4 1 4d 4 40. y − e2x cos 2 x
x 2 1 x 1 1
41. y −Î x 2 1
x 4 1 1
42. y − sx e x2 2x
sx 1 1d 2y3
43. y − x x 44. y − x cos x
45. y − x sin x 46. y − sx
x
47. y − scos xd x
48. y − ssin xd ln x
49. y − stan xd 1yx 50. y − sln xd cos x
51. Find y9 if y − lnsx 2 1 y 2 d.
52. Find y9 if x y − y x .
53. Find a formula for f snd sxd if f sxd − lnsx 2 1d.
54. Find d 9
dx 9 sx 8 ln xd.
55. Use the definition of derivative to prove that
56. Show that lim
lns1 1 xd
lim − 1
x l 0 x
1 x n
n l `S1
nD
− e x for any x . 0.
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.