EXERCISES 7.7 Values and Identities 9. (sinh x + cosh x)4 Each of Exercises 1-4 gives a value of sinh x or cosh x. Use the defi- 10. In (cosh x + sinh x) + In (cosh x - sinh x) nitions and the identity cosh x - sinh x = 1 to find the values of the remaining five hyperbolic functions. 11. Prove the identities sinh (x + y) = sinh x cosh y + cosh x sinh y, 1. sinh x = - 2. sinh x = WIP cosh (x + y) = cosh x cosh y + sinh x sinh y. 3. cosh x = 17 13 Then use them to show that 15' x 20 4. cosh x 5' x 20 a. sinh 2x = 2 sinh x cosh x. Rewrite the expressions in Exercises 5-10 in terms of exponentials and simplify the results as much as you can. b. cosh 2x = cosh x + sinh x. 5. 2 cosh (In x) 6. sinh (2 In x) 12. Use the definitions of cosh x and sinh x to show that 7. cosh 5x + sinh 5x 8. cosh 3x - sinh 3x cosh x - sinh x = 1.Finding Derivatives In Exercises 13-24, find the derivative of y with respect to the appro- priate variable. 13. y = 6 sinh ? 14. y = - sinh (2x + 1) 15. y = 2Vt tanh Vt 16. y = 12 tanh - 17. y = In (sinh z) 18. y = In (cosh z) 19. y = (sech 0)(1 - In sech 0) 20. y = (csch 0)(1 - In csch 0) 21. y = In cosh v - ~tanh2 v 22. y = In sinh v - - coth v 23. y = (x2 + 1) sech (In x) (Hint: Before differentiating, express in terms of exponentials and simplify.) 24. y = (4x2 - 1) csch (In 2x) In Exercises 25-36, find the derivative of y with respect to the appro- priate variable. 25. y = sinh ' Vx 26. y = cosh-12V x + 1 27. y = (1 - 0) tanh 10 28. y = (02 + 20) tanh '(0 + 1) 29. y = (1 - t) coth 1 Vt 30. y = (1 - t2) coth 1 t 31. y = cos x - x sech x 32. y = Inx + V1 - x2 sech- 1x 33. y = csch- NI 34. y = csch-1 20 35. y = sinh" (tan x) 36. y = cosh ' (sec x), 0