Euler's number e makes our lives a little easier since the slope of f(x)=e^x at x=0 is exactly one. In other words, e is a base such that
f'(0)=lim_{h to 0}{e^(0+h}-e^0}/h=lim_{h to 0}{e^h-1}/h=1,
which is quite useful in finding f'(x). Let us use the definition to find the derivative of f(x).
f'(x)=lim_{h to 0}{e^{x+h}-e^x}/{h}=lim_{h to 0}{e^x cdot e^h-e^x}/h
by pulling e^x out of the limit,
=e^x lim_{h to 0}{e^h-1}/h=e^x cdot 1=e^x.
Hence, the derivative of e^x is itself, which is very convenient.
I hope that this was helpful.
