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.
Here is the handout from a talk I gave on deriving the hyperbolic trig functions—this is actually a packet guiding a student through the derivation. More or less, it starts with the circular trig functions, shifts the definition to depend on area rather than arc length, constructs the comparable definition in terms of a unit hyperbola, and then bashes through some calculus to get a simpler formula, which is what you're after.