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CHAPTER 3 β€’ DERIVATIVES SECTION 3.4 EXERCISES Getting Started 1. How do you find the derivative of the product of two functions that are differentiable at a point? 2. How do you find the derivative of the quotient of two functions that are differentiable at a point? 3. Use the Product Rule to evaluate and simplify d/dx((x + 1)(3x + 2)). 4. Use the Product Rule to find f'(1) given that f(x) = x^4e^x. 5. Use the Quotient Rule to evaluate and simplify d/dx(x - 1 / 3x + 2). 6. Use the Quotient Rule to find g'(1) given that g(x) = x^2 / x + 1. 7–14. Find the derivative the following ways: a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result. b. By expanding the product first (Exercises 7–10) or by simplifying the quotient first (Exercises 11–14). Verify that your answer agrees with part (a). 7. f(x) = x(x - 1) 8. g(t) = (t + 1)(t^2 - t + 1) 9. f(x) = (x - 1)(3x + 4) 10. h(z) = (z^3 + 4z^2 + z)(z - 1) 11. f(w) = w^3 - w / w 12. g(s) = 4s^3 - 8s^2 + 4s / 4s 13. y = x^2 - a^2 / x - a, where a is a constant 14. y = x^2 - 2ax + a^2 / x - a, where a is a constant 15. Given that f(1) = 5, f'(1) = 4, g(1) = 2, and g'(1) = 3, find d/dx(f(x)g(x))|_{x=1} and d/dx(f(x)/g(x))|_{x=1}. 16. Show two ways to differentiate f(x) = 1/x^10. 17. Find the slope of the line tangent to the graph of f(x) = x / x + 6 at the point (3, 1/3) and at (-2, -1/2). 18. Find the slope of the graph of f(x) = 2 + xe^x at the point (0, 2). Practice Exercises 19–60. Derivatives Find and simplify the derivative of the following functions. 19. f(x) = 3x^4(2x^2 - 1) 20. g(x) = 6x - 2xe^x 21. f(x) = x / x + 1 22. f(x) = x^3 - 4x^2 + x / x - 2 23. f(t) = t^{5/3}e^t 24. g(w) = e^w(5w^2 + 3w + 1) 25. f(x) = e^x / e^x + 1 26. f(x) = 2e^x - 1 / 2e^x + 1 27. f(x) = xe^{-x} 28. f(x) = e^x ?x 29. y = (3t - 1)(2t - 2)^{-1} 30. h(w) = w^2 - 1 / w^2 + 1 31. h(x) = (x - 1)(x^3 + x^2 + x + 1) 32. f(x) = (1 + 1/x^2)(x^2 + 1) 33. g(w) = e^w(w^3 - 1) 34. s(t) = t^{4/3} / e^t 35. f(t) = e^t(t^2 - 2t + 2) 36. f(x) = e^x(x^3 - 3x^2 + 6x - 6) 37. g(x) = e^x / x^2 - 1 38. y = (2???x - 1)(4x + 1)^{-1} 39. f(x) = 3x^{-9} 40. y = 4 / p^3 41. g(t) = 3t^2 + 6 / t^3 42. y = w^4 + 5w^2 + w / w^2 43. g(t) = t^3 + 3t^2 + t / t^3 44. p(x) = 4x^3 + 3x + 1 / 2x^5 45. g(x) = (x + 1)e^x / x - 2 46. h(x) = (x - 1)(2x^2 - 1) / (x^3 - 1) 47. h(x) = xe^x / x + 1 48. h(x) = x + 1 / x^2e^x 49. g(w) = ???w + w / ???w - w 50. f(x) = 4 - x^2 / x - 2 51. h(w) = w^{5/3} / w^{5/3} + 1 52. g(x) = x^{4/3} - 1 / x^{4/3} + 1 53. f(x) = 4x^2 - 2x / 5x + 1 54. f(z) = (z^2 + 1 / z)e^z 55. h(r) = 2 - r - ???r / r + 1 56. y = x - a / ???x - ???a, where a is a positive constant 57. h(x) = (5x^7 + 5x)(6x^3 + 3x^2 + 3) 58. s(t) = (t + 1)(t + 2)(t + 3) 59. f(x) = ???(e^{2x} + 8x^2e^x + 16x^4) (Hint: Factor the function under the square root first.) 60. g(x) = e^{2x} - 1 / e^x - 1 61–64. Equations of tangent lines a. Find an equation of the line tangent to the given curve at a. b. Use a graphing utility to graph the curve and the tangent line on the same set of axes. 61. y = x + 5 / x - 1; a = 3 62. y = 2x^2 / 3x - 1; a = 1 63. y = 1 + 2x + xe^x; a = 0 64. y = e^x / x; a = 1 

          CHAPTER 3 β€’ DERIVATIVES
SECTION 3.4 EXERCISES
Getting Started
1. How do you find the derivative of the product of two functions that are differentiable at a point?
2. How do you find the derivative of the quotient of two functions that are differentiable at a point?
3. Use the Product Rule to evaluate and simplify d/dx((x + 1)(3x + 2)).
4. Use the Product Rule to find f'(1) given that f(x) = x^4e^x.
5. Use the Quotient Rule to evaluate and simplify d/dx(x - 1 / 3x + 2).
6. Use the Quotient Rule to find g'(1) given that g(x) = x^2 / x + 1.
7–14. Find the derivative the following ways:
a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.
b. By expanding the product first (Exercises 7–10) or by simplifying the quotient first (Exercises 11–14). Verify that your answer agrees with part (a).
7. f(x) = x(x - 1)
8. g(t) = (t + 1)(t^2 - t + 1)
9. f(x) = (x - 1)(3x + 4)
10. h(z) = (z^3 + 4z^2 + z)(z - 1)
11. f(w) = w^3 - w / w
12. g(s) = 4s^3 - 8s^2 + 4s / 4s
13. y = x^2 - a^2 / x - a, where a is a constant
14. y = x^2 - 2ax + a^2 / x - a, where a is a constant
15. Given that f(1) = 5, f'(1) = 4, g(1) = 2, and g'(1) = 3, find d/dx(f(x)g(x))|_{x=1} and d/dx(f(x)/g(x))|_{x=1}.
16. Show two ways to differentiate f(x) = 1/x^10.
17. Find the slope of the line tangent to the graph of f(x) = x / x + 6 at the point (3, 1/3) and at (-2, -1/2).
18. Find the slope of the graph of f(x) = 2 + xe^x at the point (0, 2).
Practice Exercises
19–60. Derivatives Find and simplify the derivative of the following functions.
19. f(x) = 3x^4(2x^2 - 1)
20. g(x) = 6x - 2xe^x
21. f(x) = x / x + 1
22. f(x) = x^3 - 4x^2 + x / x - 2
23. f(t) = t^{5/3}e^t
24. g(w) = e^w(5w^2 + 3w + 1)
25. f(x) = e^x / e^x + 1
26. f(x) = 2e^x - 1 / 2e^x + 1
27. f(x) = xe^{-x}
28. f(x) = e^x ?x
29. y = (3t - 1)(2t - 2)^{-1}
30. h(w) = w^2 - 1 / w^2 + 1
31. h(x) = (x - 1)(x^3 + x^2 + x + 1)
32. f(x) = (1 + 1/x^2)(x^2 + 1)
33. g(w) = e^w(w^3 - 1)
34. s(t) = t^{4/3} / e^t
35. f(t) = e^t(t^2 - 2t + 2)
36. f(x) = e^x(x^3 - 3x^2 + 6x - 6)
37. g(x) = e^x / x^2 - 1
38. y = (2???x - 1)(4x + 1)^{-1}
39. f(x) = 3x^{-9}
40. y = 4 / p^3
41. g(t) = 3t^2 + 6 / t^3
42. y = w^4 + 5w^2 + w / w^2
43. g(t) = t^3 + 3t^2 + t / t^3
44. p(x) = 4x^3 + 3x + 1 / 2x^5
45. g(x) = (x + 1)e^x / x - 2
46. h(x) = (x - 1)(2x^2 - 1) / (x^3 - 1)
47. h(x) = xe^x / x + 1
48. h(x) = x + 1 / x^2e^x
49. g(w) = ???w + w / ???w - w
50. f(x) = 4 - x^2 / x - 2
51. h(w) = w^{5/3} / w^{5/3} + 1
52. g(x) = x^{4/3} - 1 / x^{4/3} + 1
53. f(x) = 4x^2 - 2x / 5x + 1
54. f(z) = (z^2 + 1 / z)e^z
55. h(r) = 2 - r - ???r / r + 1
56. y = x - a / ???x - ???a, where a is a positive constant
57. h(x) = (5x^7 + 5x)(6x^3 + 3x^2 + 3)
58. s(t) = (t + 1)(t + 2)(t + 3)
59. f(x) = ???(e^{2x} + 8x^2e^x + 16x^4) (Hint: Factor the function under the square root first.)
60. g(x) = e^{2x} - 1 / e^x - 1
61–64. Equations of tangent lines
a. Find an equation of the line tangent to the given curve at a.
b. Use a graphing utility to graph the curve and the tangent line on the same set of axes.
61. y = x + 5 / x - 1; a = 3
62. y = 2x^2 / 3x - 1; a = 1
63. y = 1 + 2x + xe^x; a = 0
64. y = e^x / x; a = 1
Show more…
CHAPTER 3 β€’ DERIVATIVES SECTION 3.4 EXERCISES Getting Started 1. How do you find the derivative of the product of two functions that are differentiable at a point? 2. How do you find the derivative of the quotient of two functions that are differentiable at a point? 3. Use the Product Rule to evaluate and simplify d/dx((x + 1)(3x + 2)). 4. Use the Product Rule to find f'(1) given that f(x) = x^4e^x. 5. Use the Quotient Rule to evaluate and simplify d/dx(x - 1 / 3x + 2). 6. Use the Quotient Rule to find g'(1) given that g(x) = x^2 / x + 1. 7–14. Find the derivative the following ways: a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result. b. By expanding the product first (Exercises 7–10) or by simplifying the quotient first (Exercises 11–14). Verify that your answer agrees with part (a). 7. f(x) = x(x - 1) 8. g(t) = (t + 1)(t^2 - t + 1) 9. f(x) = (x - 1)(3x + 4) 10. h(z) = (z^3 + 4z^2 + z)(z - 1) 11. f(w) = w^3 - w / w 12. g(s) = 4s^3 - 8s^2 + 4s / 4s 13. y = x^2 - a^2 / x - a, where a is a constant 14. y = x^2 - 2ax + a^2 / x - a, where a is a constant 15. Given that f(1) = 5, f'(1) = 4, g(1) = 2, and g'(1) = 3, find d/dx(f(x)g(x))|x=1 and d/dx(f(x)/g(x))|x=1. 16. Show two ways to differentiate f(x) = 1/x^10. 17. Find the slope of the line tangent to the graph of f(x) = x / x + 6 at the point (3, 1/3) and at (-2, -1/2). 18. Find the slope of the graph of f(x) = 2 + xe^x at the point (0, 2). Practice Exercises 19–60. Derivatives Find and simplify the derivative of the following functions. 19. f(x) = 3x^4(2x^2 - 1) 20. g(x) = 6x - 2xe^x 21. f(x) = x / x + 1 22. f(x) = x^3 - 4x^2 + x / x - 2 23. f(t) = t^5/3e^t 24. g(w) = e^w(5w^2 + 3w + 1) 25. f(x) = e^x / e^x + 1 26. f(x) = 2e^x - 1 / 2e^x + 1 27. f(x) = xe^-x 28. f(x) = e^x ?x 29. y = (3t - 1)(2t - 2)^-1 30. h(w) = w^2 - 1 / w^2 + 1 31. h(x) = (x - 1)(x^3 + x^2 + x + 1) 32. f(x) = (1 + 1/x^2)(x^2 + 1) 33. g(w) = e^w(w^3 - 1) 34. s(t) = t^4/3 / e^t 35. f(t) = e^t(t^2 - 2t + 2) 36. f(x) = e^x(x^3 - 3x^2 + 6x - 6) 37. g(x) = e^x / x^2 - 1 38. y = (2???x - 1)(4x + 1)^-1 39. f(x) = 3x^-9 40. y = 4 / p^3 41. g(t) = 3t^2 + 6 / t^3 42. y = w^4 + 5w^2 + w / w^2 43. g(t) = t^3 + 3t^2 + t / t^3 44. p(x) = 4x^3 + 3x + 1 / 2x^5 45. g(x) = (x + 1)e^x / x - 2 46. h(x) = (x - 1)(2x^2 - 1) / (x^3 - 1) 47. h(x) = xe^x / x + 1 48. h(x) = x + 1 / x^2e^x 49. g(w) = ???w + w / ???w - w 50. f(x) = 4 - x^2 / x - 2 51. h(w) = w^5/3 / w^5/3 + 1 52. g(x) = x^4/3 - 1 / x^4/3 + 1 53. f(x) = 4x^2 - 2x / 5x + 1 54. f(z) = (z^2 + 1 / z)e^z 55. h(r) = 2 - r - ???r / r + 1 56. y = x - a / ???x - ???a, where a is a positive constant 57. h(x) = (5x^7 + 5x)(6x^3 + 3x^2 + 3) 58. s(t) = (t + 1)(t + 2)(t + 3) 59. f(x) = ???(e^2x + 8x^2e^x + 16x^4) (Hint: Factor the function under the square root first.) 60. g(x) = e^2x - 1 / e^x - 1 61–64. Equations of tangent lines a. Find an equation of the line tangent to the given curve at a. b. Use a graphing utility to graph the curve and the tangent line on the same set of axes. 61. y = x + 5 / x - 1; a = 3 62. y = 2x^2 / 3x - 1; a = 1 63. y = 1 + 2x + xe^x; a = 0 64. y = e^x / x; a = 1

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Calculus: Early Transcendentals
James Stewart 8th Edition
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In Exercises 1 through 12, use the chain rule to compute the derivative dy/dx and simplify your answer. 1. y = u^2 + 1; u = 3x - 2 2. y = 1 - 3u^2; u = 3 - 2x 3. y = sqrt(u); u = x^2 + 2x - 3 4. y = 2u^2 - u + 5; u = 1 - x^2 5. y = 1/u^2; u = x^2 + 1 6. y = 1/u; u = 3x^2 + 5 7. y = 1/(u-1); u = x^2 8. y = 1/sqrt(u); u = x^2 - 9 9. y = u^2 + 2u - 3; u = sqrt(x) 10. y = u^3 + u; u = 1/sqrt(x) 11. y = u^2 + u - 2; u = 1/x 12. y = u^2; u = 1/(x-1) In Exercises 13 through 20, use the chain rule to compute the derivative dy/dx for the given value of x. 13. y = u^2 - u; u = 4x + 3 for x = 0 14. y = u + 1/u; u = 5 - 2x for x = 0 15. y = 3u^4 - 4u + 5; u = x^3 - 2x - 5 for x = 2 16. y = u^5 - 3u^2 + 6u - 5; u = x^2 - 1 for x = 1 17. y = sqrt(u); u = x^2 - 2x + 6 for x = 3 18. y = 3u^2 - 6u + 2; u = 1/x^2 for x = 1/3 19. y = 1/u; u = 3 - 1/x^2 for x = 1/2 20. y = 1/(u+1); u = x^3 - 2x + 5 for x = 0 In Exercises 21 through 38, differentiate the given function and simplify your answer. 21. f(x) = (2x + 3)^1.4 22. f(x) = 1/sqrt(5-3x) 23. f(x) = (2x + 1)^4 24. f(x) = sqrt(5x^6 - 12) 25. f(x) = (x^5 - 4x^3 - 7)^8 26. f(t) = (3t^4 - 7t^2 + 9)^5 27. f(t) = 1/(5t^2 - 6t + 2) 28. f(x) = 2/(6x^2 + 5x + 1)^2 29. g(x) = 1/sqrt(4x^2 + 1) 30. f(s) = 1/sqrt(5s^3 + 2) 31. f(x) = 3/(1-x^2)^4 32. f(x) = 2/(3(5x^4 + 1)^2) 33. h(s) = (1 + sqrt(3s))^5 34. g(x) = sqrt(1 + 1/(3x)) 35. f(x) = (x + 2)^3(2x - 1)^5 36. f(x) = 2(3x + 1)^4(5x - 3)^2 37. f(x) = (x + 1)^5/(1 - x)^4 38. f(x) = (1 - 5x^2)/cbrt(3 + 2x)

Adi S.
h-gw-continuity-ol-fu-4-lim-fx-h-fl-4us-8dx-follows-that-exercises-23-exenises-through-1-differentiate-the-gwen-ficiion-fw-is-ft-s1-i3u-1xi-40j15-x93t-x-1-47-41-4-22-5-2r-y-gvs-x2-r-o-in-exe-56233

lim_{h o 0} frac{g(x+h) - g(x)}{h} = frac{dg}{dx} lim_{h o 0} f(x+h) = f(x) frac{d}{dx}(fg) = f frac{dg}{dx} + g frac{df}{dx} EXERCISES 2.3 In Exercises 1 through 18, differentiate the given function. 1. f(x) = (2x + 1)(3x - 2) 2. f(x) = (x - 5)(1 - 2x) 3. y = 10(3u + 1)(1 - 5u) 4. y = 400(15 - x^2)(3x - 2) 5. f(x) = frac{1}{3}(x^5 - 2x^3 + 1)(x - frac{1}{x}) 6. f(x) = -3(5x^3 - 2x + 5)(sqrt{x} + 2x) 7. y = frac{x + 1}{x - 2} 8. y = frac{2x - 3}{5x + 4} 9. f(t) = frac{t}{t^2 - 2} 10. f(x) = frac{1}{x - 2} 11. y = frac{3}{x + 5} 12. y = frac{t^2 + 1}{1 - t^2} 13. f(x) = frac{x^2 - 3x + 2}{2x^2 + 5x - 1} 14. g(x) = frac{(x^2 + x + 1)(4 - x)}{2x - 1} 15. f(x) = (2 + 5x)^2 16. f(x) = (x + frac{1}{x})^2 17. g(t) = frac{t^2 + sqrt{t}}{2t + 5} 18. h(x) = frac{x}{x^2 - 1} + frac{4 - x}{x^2 + 1} In Exercises 19 through 23, find an equation for the tangent line to the given curve at the point where x = x_0. 19. y = (5x - 1)(4 + 3x); x_0 = 0 20. y = (x^2 + 3x - 1)(2 - x); x_0 = 1 21. y = frac{x}{2x + 3}; x_0 = -1 22. y = frac{x + 7}{5 - 2x}; x_0 = 0 23. y = (3sqrt{x} + x)(2 - x^2); x_0 = 1 In Exercises 24 through 27, find all points on the graph of the given function where the tangent line is horizontal. 24. f(x) = (x - 1)(x^2 - 8x + 7) 25. f(x) = (x + 1)(x^2 - x - 2) 26. f(x) = frac{x^2 + x - 1}{x^2 - x + 1} 27. f(x) = frac{x + 1}{x^2 + x + 1} In Exercises 28 through 31, find the rate of change of y for the prescribed value of x_0. 28. y = (x^2 + 2)(x + sqrt{x}); x_0 = 4 29. y = (x^2 + 3)(5 - 2x^3); x_0 = 1 30. y = frac{2x - 1}{3x + 5}; x_0 = 1 31. y = x + frac{3}{2 - 4x}; x_0 = 0 The normal line to the curve y = f(x) at the point P with coordinates (x_0, f(x_0)) is the line perpendicular to the tangent line at P. In Exercises 32 through 33, find an equation for the normal line to the given curve through the prescribed point. 32. y = x^2 + 3x - 5; (0, -5) 33. y = frac{2}{x} - sqrt{x}; (1, 1)

Adi S.
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00:01 Resolutions per the equivalent for the question.
00:03 In this question given there, apox is called x x 4, multiple by u.
00:08 Power x, is equal to u .s...
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