TITLE: Non-standard analysis way of proving that derivative of $e^x$ is $e^x$
QUESTION [7 upvotes]: What is the non-standard (infinitesimal) analysis way of proving that the derivative of $e^x$ is $e^x$? I tried to prove it myself, but I am having a hard time proving this without recourse to standard limit things.

REPLY [3 votes]: $$
\frac{e^{x + dx} -e^{x}}{dx}=  e^x \left( \frac{e^{dx} -1}{dx} \right)
$$
You transfer everything to non-standrad reals, so that $dx$ could become infinitesimal. 
Then you evaluate the exponential from its infinite series
$$
 \frac{e^{dx} -1}{dx} = 1+  \frac{dx}{2} + H.O.T (dx^2)
$$
the standard part of the last one  is  1.
It is exactly the same technique like working with approximations.