Hyperbolic functions are similar to trigonometric functions

Question

Hyperbolic functions are similar to trigonometric functions. Define hyperbolic cosine, denoted cosh(x) as

cosh(x) = (e^x + e^-x) / 2,

and define hyperbolic sine, denoted sinh(x), as

sinh(x) = (e^x + e^-x) / 2,

1. Use the definition of cosh(x) and sinh(x) to show that

(cosh(x))² - (sinh(x))² = 1.

2. The inverse cosh(x) and sinh(x) arc denoted by arccosh(x) and arcsinh(x). Use implicit differentiation to compute (d/dx) arccosh(x) and (d/dx) arcsinh(x).

3. Use the definition of cosh(x) to verify that

arccosh(x) = In (x + √(x² - 1)

4. Use the definition of arccosh(x) from 3 to compute (d/dx) arccosh(x). Does this match your answer from 2? Explain.