SECTION 3.6Derivatives of Logarithmic Functions2233.6 EXERCISES1. Explain why the natural logarithmic functiony In x is usedmuch more frequently in calculus than the other logarithmicfunctions y33-34 Find an equation of the tangent line to the curve at thegiven point.log,x.33. y In(x 3x +1), (3,0)2-22 Differentiate the function.34. y x2 In x, (1,0)2. f(x)=x In x- x3. f(x)= sin( In x)A35. If f(x) = sin x + In x, find f'(x). Check that your answer isreasonable by comparing the graphs of f andf'.4. f(x)In(sinx)5. f(x)= In16. уX36. Find equations of the tangent lines to the curve y = (In x)/xIn xat the points (1,0) and (e, 1/e). Illustrate by graphing thecurve and its tangent lines.7. f(x)log 10(1 cos x)8. f(x) log10Vx9. g(x)In(xe 2x)37. Let f(x)=f'(T/4) 6?=cx +In(cos x). For what value of c is10. g(t) 1 +Int11. F(t)=(In t) sin t12. h(x) In(x +x21)3?38. Let f(x)= log,(3x2 - 2). For what value of b is f'(1)(2y1)Vy2 139-50 Use logarithmic differentiation to find the derivative of thefunction.13. G(y) InIn v14. P(v)1-e cos x40. у 339. y (x22)(x4)15. F(s) In ln s16. y In 1 + t - t'|х17. T(z)42. y xe(x + 1)22 log2zcot x)18. y In(csc x -41. yx4 1a2 z220. H(z)=In z244. y x43. y x19. y ln(e* + xe_*)46. y (x)45. y xsinxlog2 (x logs x)(sin x)n21. y tan[In(ax + b)]22. y48. y47. y (cos x)*50. y (In x)os49. y (tan x)1/x23-26 Find y' and y"In x24. yV In x51. Find y' if y ln(x2 + y2).23. у 31 + In x52. Find y' if x = y.26. y In(1 + In x)25. y In sec x|53. Find a formula for f((x) if f(x) = In(x - 1).d9(x8 In x)dx27-30 Differentiate f and find the domain of f.54. Find28. f(x) 2 + In xX27. f(x)1 - In(x 1)55. Use the definition of derivative to prove thatIn(1+ x)lim30. f(x) In In In x29. f(x) In(x2 2x)X(0-)-.= ex for any X31. If f(x) In(x + In x), find f'(1).56. Show that lim 1cos (In x2), find f'(1).32. If f(x)

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Asked Oct 15, 2019

I need help with question 29 in Section 3.6, page 223, of the James Stewart Calculus Eighth Edition textbook.

SECTION 3.6
Derivatives of Logarithmic Functions
223
3.6 EXERCISES
1. Explain why the natural logarithmic functiony In x is used
much more frequently in calculus than the other logarithmic
functions y
33-34 Find an equation of the tangent line to the curve at the
given point.
log,x.
33. y In(x 3x +1), (3,0)
2-22 Differentiate the function.
34. y x2 In x, (1,0)
2. f(x)=x In x- x
3. f(x)= sin( In x)
A35. If f(x) = sin x + In x, find f'(x). Check that your answer is
reasonable by comparing the graphs of f andf'.
4. f(x)In(sinx)
5. f(x)= In
1
6. у
X
36. Find equations of the tangent lines to the curve y = (In x)/x
In x
at the points (1,0) and (e, 1/e). Illustrate by graphing the
curve and its tangent lines.
7. f(x)
log 10(1 cos x)
8. f(x) log10Vx
9. g(x)In(xe 2x)
37. Let f(x)=
f'(T/4) 6?
=cx +In(cos x). For what value of c is
10. g(t) 1 +Int
11. F(t)=(In t) sin t
12. h(x) In(x +x21)
3?
38. Let f(x)= log,(3x2 - 2). For what value of b is f'(1)
(2y1)
Vy2 1
39-50 Use logarithmic differentiation to find the derivative of the
function.
13. G(y) In
In v
14. P(v)
1-
e cos x
40. у 3
39. y (x22)(x4)
15. F(s) In ln s
16. y In 1 + t - t'|
х
17. T(z)
42. y xe(x + 1)
22 log2z
cot x)
18. y In(csc x -
41. y
x4 1
a2 z2
20. H(z)=In z2
44. y x
43. y x
19. y ln(e* + xe_*)
46. y (x)
45. y xsinx
log2 (x logs x)
(sin x)n
21. y tan[In(ax + b)]
22. y
48. y
47. y (cos x)*
50. y (In x)os
49. y (tan x)1/x
23-26 Find y' and y"
In x
24. y
V In x
51. Find y' if y ln(x2 + y2).
23. у 3
1 + In x
52. Find y' if x = y.
26. y In(1 + In x)
25. y In sec x|
53. Find a formula for f((x) if f(x) = In(x - 1).
d9
(x8 In x)
dx
27-30 Differentiate f and find the domain of f.
54. Find
28. f(x) 2 + In x
X
27. f(x)
1 - In(x 1)
55. Use the definition of derivative to prove that
In(1+ x)
lim
30. f(x) In In In x
29. f(x) In(x2 2x)
X
(0-)-.
= ex for any X
31. If f(x) In(x + In x), find f'(1).
56. Show that lim 1
cos (In x2), find f'(1).
32. If f(x)
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SECTION 3.6 Derivatives of Logarithmic Functions 223 3.6 EXERCISES 1. Explain why the natural logarithmic functiony In x is used much more frequently in calculus than the other logarithmic functions y 33-34 Find an equation of the tangent line to the curve at the given point. log,x. 33. y In(x 3x +1), (3,0) 2-22 Differentiate the function. 34. y x2 In x, (1,0) 2. f(x)=x In x- x 3. f(x)= sin( In x) A35. If f(x) = sin x + In x, find f'(x). Check that your answer is reasonable by comparing the graphs of f andf'. 4. f(x)In(sinx) 5. f(x)= In 1 6. у X 36. Find equations of the tangent lines to the curve y = (In x)/x In x at the points (1,0) and (e, 1/e). Illustrate by graphing the curve and its tangent lines. 7. f(x) log 10(1 cos x) 8. f(x) log10Vx 9. g(x)In(xe 2x) 37. Let f(x)= f'(T/4) 6? =cx +In(cos x). For what value of c is 10. g(t) 1 +Int 11. F(t)=(In t) sin t 12. h(x) In(x +x21) 3? 38. Let f(x)= log,(3x2 - 2). For what value of b is f'(1) (2y1) Vy2 1 39-50 Use logarithmic differentiation to find the derivative of the function. 13. G(y) In In v 14. P(v) 1- e cos x 40. у 3 39. y (x22)(x4) 15. F(s) In ln s 16. y In 1 + t - t'| х 17. T(z) 42. y xe(x + 1) 22 log2z cot x) 18. y In(csc x - 41. y x4 1 a2 z2 20. H(z)=In z2 44. y x 43. y x 19. y ln(e* + xe_*) 46. y (x) 45. y xsinx log2 (x logs x) (sin x)n 21. y tan[In(ax + b)] 22. y 48. y 47. y (cos x)* 50. y (In x)os 49. y (tan x)1/x 23-26 Find y' and y" In x 24. y V In x 51. Find y' if y ln(x2 + y2). 23. у 3 1 + In x 52. Find y' if x = y. 26. y In(1 + In x) 25. y In sec x| 53. Find a formula for f((x) if f(x) = In(x - 1). d9 (x8 In x) dx 27-30 Differentiate f and find the domain of f. 54. Find 28. f(x) 2 + In x X 27. f(x) 1 - In(x 1) 55. Use the definition of derivative to prove that In(1+ x) lim 30. f(x) In In In x 29. f(x) In(x2 2x) X (0-)-. = ex for any X 31. If f(x) In(x + In x), find f'(1). 56. Show that lim 1 cos (In x2), find f'(1). 32. If f(x)

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check_circleExpert Solution
Step 1
  • Recall d[ln(x)] / dx = 1/x
  • Recall the chain rule of differentiation
Step 2

Please see the white board. The expression for f'(x) is the derivative.

2х — 2
f(ar) In(a2-2r) f'(x)
а2 — 2т
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2х — 2 f(ar) In(a2-2r) f'(x) а2 — 2т

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Step 3

ln(x) is defined for x>0. Log of a negative number is not defined.

Hence, ln(x2 – 2x) is ...

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