# 3.6

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1. Prove the identity. Cosh(-x) = cosh x (This shows that cosh is an even function.)
2. Prove the identity. Cosh x + sinh x = e^x cosh x + sin x = 1/2 (e^x + e^-x)
3. Prove the identity. Sinh 2x = 2 sinh x cosh x
Sinh 2x = sinh(x+ )
4. ( 1+ tanh x )/(1-tanh x) = e^2x
5. If tanh x = 4/5, find the values of the other hyperbolic functions at x.
6. Prove the formulas given in this table for the derivates of the functions cosh, tanh , csch, sech, coth. Which of the following are proven correctly? (Select all that apply.)
7. Find the derivative. Simplify where possible. F(x) = tanh(5 + e^5x)
8. Find the derivative. Simplify where possible. F(x) = x sinh x – 9 cosh x
9. Find the derivative. Simplify where possible. G(x) = cosh(ln x)
10. Find the derivative. Simplify where possible. H(x) = ln(cosh(9x))
11. Find the derivative. Simplify where possible. Y = x coth(7 + x^2)
12. Find the derivative. Simplify where possible. Y = e^cosh(4x)
13. Find the derivative. Simplify where possible. F(t) = sech^2(5e^t)
14. Find the derivative. Simplify where possible. G(x) = (5 – cosh x)/(5 + cosh x)
15. Find the derivative. Simplify where possible. Y = x arc tanh x + ln(1 – x^2)^1/2
16. Find the derivative. Simplify where possible. Y = x arc sinh(x/7) – (49 + x^2)^1/2
17. Show that d/dx arctan(tanh x) = sech 2x
18. Find the derivative. Simplify where possible. Y = arctanh (x)^1/2
19. Find the derivative. Simplify where possible. Arccoth(x^2 + 2)^1/2
20. If cosh x = 5/3 and x> 0, and find the values of the other hyperbolic functions at x.