Is the natural exponential function defined as being its own derivative? Is $e^x$ actually defined as being the function $f$ for which $\dfrac{d}{dx}f=f$?
By which I mean not "does the identity hold", of course I know it does and that this definition is sufficient for $e$, but did Euler actually sit down and think "gee, I wonder what I can differentiate to get the same thing back"?
I guess my question is equivalent to "what was the first use of Euler's constant" or "why did Euler come up with [what we call] $e$".

Gregoire de Saint-Vincent and Alphonse Antonio de Sarasa  around 1690 studied the question of how to compute areas under the hyperbola $xy=1$, which led to the notion of how to compute the area under the curve $y=1/x$. While the case $y=1/x^n$, $n > 1$ was simpler and solved by Cavalieri earlier, a new function had to be defined for the case $n=1$. They introduced this notion of a "hyperbolic logarithm".
Euler, about 40 years later, introduced $e$ as the constant which gave area $1$ in a letter to Goldbach.
The limit expression $\lim_{n\to\infty} (1+\frac{1}{n})^n$ was introduced by Bernoulli even earlier than this, and I'm not entirely sure when the notions were found to coincide.
Source: https://en.wikipedia.org/wiki/E_(mathematical_constant)#History
