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I know that we can differentiate $e^x$ by using limits. But why is it that it is the derivative of it's own? Or is it the case that there must exist a function which should be it's own derivative and we have defined it to be $e^x$?

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    How were you introduced to number $e$? – 2017-01-24
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    Yes, the initial value problem $\frac{dy}{dx}=y$ with $y(0) = 1$ has a unique solution for $x\ge 0$. The exponential function, like all good things, has many definitions though and you takes your pick where to start. – 2017-01-24
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    The latter statement is correct. It is the unique solution to $f' =f$. – 2017-01-24
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    @imranfat I was first introuduced $e$ as $\lim_{n \to \infty}(1+\frac{1}{n})^n$ and later in calculus I was told it's power series. – 2017-01-24
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    That limit is a consequence of a lot of stuff if you introduce the natural log as an area under the curve $y=1/x$ from $x=1$ to $x=a$ as $lna$. From there you arrive (after whole bunch of steps) that base $e$ essentially is defined as the base of the exponential function $y=a^x$ such that it's derivative is the same, i.e.$lne=1$ – 2017-01-24

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There is an excellent book, "e:" The Story of a Number, that goes through the many ways $e$ and $e^x$ and $\ln(x)$ can be defined, and the historical motivations for the development of that math.

A bit surprisingly, the historical reasoning was:

  • The integral of the function $f(x) = \frac1x$ exists and $\int_1^t \frac1x\,dx$ is a single-valued function of the upper limit $t$. Let's name that function $\log(t)$.

  • You can prove some interesting things about $\log(t)$ including the familar property that $\log(uv) = \log(u)+\log(v)$ just from the definition as an integral (!)

  • You can easily prove that $\log(x)$ is monotonic increasing and continuous, so there is an inverse function, which for now we can call $\exp(x)$.

  • Using the properties of $\log(x)$ you can prove that $\exp(x)$ has the usual properties of some power $a^x$, for example, $\exp(x)\exp(y) = \exp(x+y)$.

  • If we identify the base of that power as some specific number, call it $e$, we can then prove some amazing properties of $e$, including that $$ \frac{d(e^x)}{dx} = e^x\\ e = \lim{x\to\infty} \left( 1+\frac1x\right)^x \\ e^x = \sum_{n=0}^\infty \frac{x^n}{n!} $$

The proof that $\frac{d(e^x)}{dx} = e^x$ is particularly easy, because $\frac{d(\log(x))}{dx} = \frac1x$ by the definition of $\log(x)$, and $e^x$ is the inverse function of $\log(x)$.

The book, by the way, also tells you why "log" is related to the word "log" meaning a piece of a tree. I won't spoil it by revealing the relation; you will love to read the book.

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    That's indeed the sequence of topics I was referring to. +1 – 2017-01-24