Could you explain why $\frac{d}{dx} e^x = e^x$ "intuitively"? As the title implies, It is seems that $e^x$ is the only function whoes derivative is the same as itself.
thanks.

Well, think of exponential growth (like e.g. bacteria grow):
We know, the more bacteria exist in a colony, the faster the colony will grow. More precisely: The growth speed of the colony $B$ is proportional to it's size ... Double size, double speed.
$$\frac{dB}{dt} \sim B$$
Furthermore we know the growth is exponential, since bacteria clone themselves in fixed amounts of time, i.e.
$$B \sim 2^{k\cdot t}$$
Putting it together, we can deduce that:
$$\frac{d}{dt}2^{kt} = c \cdot 2^{kt}$$
or with $a = 2^k$
$$\frac{d}{dt}a^t = c \cdot a^t$$

Now, how do we get the $e$? We just ask: What base $a$ do we have to take such that $c = 1$, i.e. $\dot{B} = B$?
We simply call that base $e$. Having such an $e$ is quite useful. We could use its special derivation traits we found above to define all exponential functions to the base $e$.
$$a^x = e^{x \cdot \ln a} $$
This shows that the factor $c$ we encountered in the above equations equals $\ln a = \log_e a$ and therefore, we can easily derive all kinds of exponential terms.
After all, $e$ turns out to be $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x \approx 2.718\ldots$
