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)
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)
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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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)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b9bfc7b-5e86-435c-b9dd-a07d54ff6a7e%2F02522261-70c8-4627-8f5e-e1f73afa0613%2Fn1o65oi.jpeg&w=3840&q=75)
Transcribed Image Text: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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