# Could you explain why ddxex=ex\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.

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 $a = 2^k$

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 $e$.

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 $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x \approx 2.718\ldots$