I know that functions which are associated with the geometry of the conic section called a hyperbola are called hyperbolic functions. But where on earth did '$e$' come from? I really don't understand. I've seen a lot of math texts where they introduce hyperbolic functions by just writing out equations of $\sinh$, $\cosh$, $\tanh$, etc., without mentioning where they came from. I am looking for a possible derivation of this. I'll be glad if someone could help me by deriving this or even refer me to some source where I can find the derivation.
$\sinh x= [e^x −e^{-x}] / 2$
Once I obtain the derivation for $\sinh x$ I'll try to figure out $\cosh$ and $\tanh$.
Thank you.