434 Chapter 7 Transcendental Functions Finding Derivatives In Exercises 13-24, find the derivative of y with respect to the appro- 47. sech- ( x - ) dx 48. csch- (5 - x) dx priate variable. 13. y = 6 sinh ~ 14. y = - sinh (2x + 1) 49. sech Vitanh Vidt 50. csch (In t) coth (In t) dt Vi 15. y = 2Vt tanh Vt 16. y = 12 tanh - , coth x dx 52. tanh 2x dx 17. y = In (sinh z) 18. y = In (cosh z) 19. y = (sech 0)(1 - In sech 0) 20. y = (csch @)(1 - In csch () In 2 53. -In 4 Ze cosh 0 de 4e * sinh 0 do 21. y = In cosh v - - tanh2 v 22. y = In sinh v - - coth v ~ TT /4 23. y = (x2 + 1) sech (In x) 55. -TT / 4 cosh (tan 0) sec2 0 de 56. 2 sinh (sin 0) cos 0 de (Hint: Before differentiating, express in terms of exponentials and simplify.) 57 . cosh (In ?) at 58. 8 cosh Vxx 24. y = (4x2 - 1) csch (In 2x) Vx 59. /m cost ( 2 ) ax n 10 In Exercises 25-36, find the derivative of y with respect to the appro- 60. priate variable. 4 sinh (2) dx 25. y = sinh ' Vx 26. y = cosh 12Vx + 1 27. y = (1 - 0) tanh 10 28. y = (02 + 20) tanh '(0 + 1) Inverse Hyperbolic Functions and Integrals 29. y = (1 - t) coth ' Vt 30. y = (1 - 2) coth Since the hyperbolic functions can be written in terms of exponential functions, it is possible to express the inverse hyperbolic functions in 31. y = cos lx - x sech x 32. y = Inx + V1 - x2 sech 'x terms of logarithms, as shown in the following table. 33. y = csch-' ) 34. y = csch-1 20 35. y = sinh" (tan x) sinh 'x = In (x + Vx2 + 1), - 00 1 38. x sech 'x dx = 12 2 sech - 1x - JV1 - x + C 39. x coth 'xdx = 2 coth 'x + 2 + c Use these formulas to express the numbers in Exercises 61-66 in terms of natural logarithms. 61. sinh '(-5/12) 62. cosh (5/3) . tanh 'x dx = xtanh 'x + , In (1 - x ) + C 63. tanh '(-1/2) 64. coth ' (5/4) 65. sech ' (3/5) 66. csch-1 (-1/ V3) Evaluating Integrals Evaluate the integrals in Exercises 41-60. Evaluate the integrals in Exercises 67-74 in terms of 41. sinh 2x dx . sinh *dx a. inverse hyperbolic functions. b. natural logarithms. 2V/3 43. 6cosh - In 3 dx 44. 4cosh (3x - In 2) dx 67. dx 6 dx V 4 + x 2 VI + 9x 45. tanh # dx 46. coth 69. 51-X 70. 1-x