What's the derivation of the hyperbolic functions?

Answer 1

That question requires at least an hour class and you should read a chapter in a calculus textbook which describes the whole derivation including a comparison of the circular radian and the hyperbolic radian.

The short answer is that cosh x = (e^x + e^-x)/2 and sinh x = (e^x - e^-x)/2 and the other 4 functions are defined exactly in the same way as the circular functions are.