Suddenly found that exponential function is very wonderful. It is the eigen-function of differential operator. because of that, it appears in almost everywhere as differential operator constantly appears.

The definition of the exponential $e$ is,

$\displaystyle e = \lim_{n\to\infty} \left(1+\frac{1}{n} \right)^n$

this is the 1% growth continually in a unit of time. It is easy to see that

$\displaystyle e^x = \lim_{n\to\infty} \left(1+\frac{x}{n} \right)^n$

$e^x e^y = e^{x+y}$

expanding,

$\displaystyle \left(1+\frac{x}{n} \right)^n = \sum_{r=0}^{n} C_r^n \left(\frac{x}{n} \right)^r = \sum_{r=0}^{n} \left(\prod_{k=0}^{r-1} \frac{n-k}{n} \right)\frac{x^r}{r!}$,

Take $n\to\infty$,

$\displaystyle e^x = \sum_{r=0}^{\infty} \frac{x^r}{r!}$

It is easy to see that

$\displaystyle \frac{d}{dx} e^x = e^x$