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Number $e$: what is the intuitive relationship between the continuolsy compound interest, $\lim_{n \to \infty} (1 + \frac{1}{n})^n,$ and that $\frac{d}{dx}e^x = e^x$ is the derivation of the number $e$?  

Let me explain: If we define $e$ as a number satisfying $\frac{d}{dx} a^x = a^x,$ namely satisfying equation $\frac{a^{x + h} - a^x}{h} = a^x \iff a = (1 + h)^{\frac{1}{h}}$ as $h \rightarrow 0$ or equivalent $\lim_{n \to \infty} (1 + \frac{1}{n})^n$ where is intutition of why this number appeared?

I wonder what the connection is between these two concepts? Why did we get this one and not another number (in intuitive sense)? Algebraically everything is correct, but I do not see a logical explanation, that is, a cause and effect relationship between these two concepts.

I have seen a lot explanations of the number $e$ (limit definition) on the internet, but no one explains the cause-and-effect relationship between that limit (continuosly compounding interest) and $\frac{d}{dx}e^x = e^x.$

In short can you explain to me why compund interest and "selfderivation" are, we can see, same concepts? Can you explain what that limit even mean?

Someone who learns
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    Continuous compound interest is what happens in a situation where the rate of increase of your dollar amount is proportional to the amount of dollars you have. In such a situation, having twice as much money means you're making money twice as fast. But the exponential functions $Pe^{rt}$ are the only functions which grow in this manner. – littleO Apr 12 '20 at 11:00
  • Thanks. Maybe some example, visual interpretation? – Someone who learns Apr 12 '20 at 12:11

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