The first time I knew about the hyperbolic function was when I was studying the derivatives. And I know that the derivative of   $\sinh x=  ( e^x - e^{-x} )/ 2$ , but i still confused with what are they really are? and how did we get them and for what we are using them?  Thanks very much, and i hope that wasn't a long question.
I covered hyperbolic trigonometric functions in a recent maths course. However I was never presented with any reasons as to why (or even if) they are useful.  Is there any good examples of their uses outside academia?
I was wondering on the following and I probably know the answer already: NO. Is there another number with similar properties as $e$? So that the derivative of $\exp(x)$ is the same as the function itself. I can guess that it's probably not, because otherwise $e$ wouldn't be that special, but is there any proof of it?