Lecture 19 : Definition of the natural exponential function [Section 19.2]
19.2.2
Theorem:
The function has the following properties:
(i)
(ii)
The function is differentiable and
(iii)
For
(iv)
The function is strictly increasing and concave upward on
(v)
Proof:
That ,follow from the fact that .
The differentiability of follows from the theorem on the derivative of the inverse function. Further, its derivative at a point is given by
This proves (ii).
To prove (iii), let
has the following properties: 
is differentiable and 
is strictly increasing and concave upward on 
,follow from the fact that 
. 
follows from the theorem on the derivative of the inverse function. Further, its derivative at a point 
is given by 
