Differentiation Rules CHAPTER 3 264 d [tanh (sin x)]. dx EXAMPLE 5 Find SOLUTION Using Table 6 and the Chain Rule, we have 1 (sin x) [tanh (sin x)] dx (sin x)2 dx 1 COSX =sec x COS x= cosx 1 sin2x 3.11 EXERCISES 21. If cosh x =and x> 0, find the values of the other hyper- bolic functions at x 1-6 Find the numerical value of each expression. (b) cosh 0 1. (a) sinh 0 22. (a) Use the graphs of sinh, cosh, and tanh in Figures 1-3 to (b) tanh 1 2. (a) tanh 0 draw the graphs of csch, sech, and coth. (b) Check the graphs that you sketched in part (a) by using a graphing device to produce them. (b) cosh 5 3. (a) cosh(In 5) 4. (a) sinh 4 (b) sinh(ln 4) 23. Use the definitions of the hyperbolic functions to find each of the following limits. (a) lim tanh x 5. (a) sech 0 (b) cosh1 (b) sinh 1 6. (a) sinh 1 (b) lim tanh x (d) lim sinh x (c) lim sinh x 7-19 Prove the identity (e) lim sech x (f) lim coth x 7. sinh(=x) (This shows that sinh is an odd function.) = -sinh x (g) lim coth x (h) lim coth x 8. cosh(=x) = cosh x (This shows that cosh is an even function.) sinh x (i) lim csch x (j) lim r 0 et 9. cosh x +sinh x = e 24. Prove the formulas given in Table 1 for the derivatives of the functions (a) cosh, (b) tanh, (c) csch, (d) sech, and (e) coth. 10. Cosh x sinh x e * 11. sinh(x+ y) = sinh x cosh y + cosh x sinh y 25. Give an alternative solution to Example 3 by letting y sinhx and then using Exercise 9 and Example 1(a) with x replaced by y. 12. cosh(xy) = cosh x coshy + sinh x sinhy 13. coth2x- 1 = csch?r 26. Prove Equation 4. tanh x + tanhy 14. tanh(x+y) 27. Prove Equation 5 using (a) the method of Example 3 and (b) Exercise 18 with x replaced by y. 1 +tanh x tanh y 15. sinh 2x2 sinh x cosh x 28. For each of the following functions (i) give a definition like those in (2), (ii) sketch the graph, and (iii) find a formula similar to Equation 3. (a) csch 16. cosh 2x =cosh2r +sinh'x x2 1 17. tanh(In x) (b) sech x21 (c) coth 1 + tanh x 18. 1 - tanh x 29. Prove the formulas given in Table 6 for the derivatives of the following functions. (a) cosh (d) sech e 2x 19. (cosh x + sinh x)" = cosh nx + sinh nx (n any real number) (b) tanh (e) coth (c) csch 30-45 Find the derivative. Simplify where possible. 20. If tanh x 1, find the values of the other hyperbolic func- 30. f(x)e cosh x tions at x. 31. f(x) tanh Vx 32. g(x) sinh2x = 2