Could you explain why ddxex=ex\frac{d}{dx} e^x = e^x “intuitively”?

As the title implies, It is seems that ex is the only function whoes derivative is the same as itself.

thanks.

Answer

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.

dBdtB

Furthermore we know the growth is exponential, since bacteria clone themselves in fixed amounts of time, i.e.

B2kt

Putting it together, we can deduce that:

ddt2kt=c2kt

or with a=2k

ddtat=cat


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.

ax=exlna

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

Attribution
Source : Link , Question Author : Jichao , Answer Author : Srivatsan

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