One way determine the deriveative of is to use inverse functions (natural logs). Another way is to use the definition of a derivative. Thus, . When simplified, this results in the equation .
My questions is: what is a proof that ?
Your question is fully justified!.. effectively the limit is a 'fundamental limit' from which You can derive all the properties of the exponential function. To demonstrate it we can start from the well known limit...
(1)
Now if You can set so that is and therefore...
(2)
Merry Christmas from Italy