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In Exercises 1-20, find the derivative. 1. y = x lnx 2. y=

In Exercises 1-20, find the derivative. 1. y = x lnx 2. y=

Calculus: Early Transcendentals
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...
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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
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
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