There is a simple reason why one rarely sees a proof of the value of the derivative of e^x in textbooks. One never starts from e=2.718... or as the value of some limit and prove things from there on. It doesn't make sense, it would be like defining π from one of the many series that converge to it (or to some number related to it). Defining e like that doesn't explain why we have given a short and convenient name to this number and not to the value of some other limit.
Another flaw is that one has to give a meaning to things like eπ. The very notion of powers is tied to the exponential function, and this is where one should start, with a function. One usually starts by defining exp(x) to be the function satisfying the differential equation y'=y and such that exp(0)=1.
If you know about Taylor series, or more generally about expanding a function as a series, you will know that we can write:
exp(x)=1+exp'(0)x + o(x)=1+x + o(x)
Details on what o(x) means are here
, but basically it means "something small compared to x".
We can integrate this series to get:
where c is a constant of integration
. One of the properties of such series is that when you add on terms, the first ones stay the same, so c=1
We can continue this, and so we have
which we can write as exp(x)=Σ∞i=0
A bit of analysis shows that this series converges everywhere, and thus we can calculate exp(x) everywhere in this manner.
Fundamental properties of the exponential
Here we will prove the 2 basic properties:
Consider the function f(x)=exp(x+y).
We have, f(0)=exp(y), and f'=f so f'(0)=exp(y). We use the same method for calculating f(x) as we did for calculating exp(x):
We do our integration
Now this is just exp(y) times the beginning of the series for exp(x). Since integration
, we can in fact pull out the exp(y) factor, and have exp(x+y)=exp(y)*exp(x)
Before you know about exponentials, it only makes sense to define integer powers.
We note that
So the 2 expressions coincide for integer
values of y. Until now we haven't defined what it meant for a number to be raised to the √2 th power for example. This means we are able to define exp(yx)=(exp(x))y
for all values of y, we extend the meaning of "raise to a power"
You may notice that the exponential function is behaving in the same way as a power function, with the base exp(1). We then have exp(x)=ex.
It is left as an exercise to the reader to deduce from this other properties of the exponential, such as its inverse function, or that exp(1)= limn→ ∞(1+1/n)n. The derivative of the exponential function being equal to the function itself is really much more of a defining feature than a property.