Why is #d/dxe^x=e^x#?
This follows from the definition of natural logarythm and its inverse.
The "why" depends on how you've defined
One approach is to define
then to define
In this case
Differentiating implicitly gets us
(We owe you a proof that this is well-defined.)
Then, using the definition of derivative:
We then define
With this definition we get
# = e^xlim_(hrarr0)((e^h-1)/h) = e^x#
(We owe you a proof that this is well defined.)
Differentiating term by term (we owe you a proof that this is possible), we get
Which simplifies to