Differentiating Hyperbolic Functions

[BoT]

1. Differentiate cosh(x) using first principles



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


From previous exercises, I know the answer will be sinh(x)= (e^x-e^-x)/2 but I cannot get to the answer.
I seem to be left with the equation: lim h ---> 0 (e^2x*e^2h +1-e^h*2e^x +e^h)/(2h*e^x*e^h)
But when you make h=0 , it becomes underfined?

Homework Statement

 

[rock.freak667]

Where did you get e^2x and whatnot from?

Just simplify

$$\frac{e^{x+h}+e^{-(x+h)}-(e^x+e^{-x})}{2h}$$


and then factorize.

 

[BoT]

I got it to

e^x*e^h+e^-x*e^-h -e^x -e^-x/ 2h

But how do I simplify this?
This is the part that always screws me up!

 

[rock.freak667]

I got it to

e^x*e^h+e^-x*e^-h -e^x -e^-x/ 2h

But how do I simplify this?
This is the part that always screws me up!

Factor out e^x from e^x*e^h and e^x. Do a similar exercise with e^-x from the terms.

Then compute the limits.