As the title implies, It is seems that ex is the only function whoes derivative is the same as itself.
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.
Furthermore we know the growth is exponential, since bacteria clone themselves in fixed amounts of time, i.e.
Putting it together, we can deduce that:
or with a=2k
Now, how do we get the e? We just ask: What base a do we have to take such that c=1, i.e. ˙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.
This shows that the factor c we encountered in the above equations equals lna=logea and therefore, we can easily derive all kinds of exponential terms.
After all, e turns out to be lim