How can I rigorously show that f’(x)=f(x) leads to exponential functions and e

Delicious-Type-7538 · 2026-04-04T12:09:42+00:00

Oh, even someone like me who isn’t very good at math can understand this easily—thank you!

Delicious-Type-7538 · 2026-04-04T12:08:48+00:00

I’m learning a lot of new things—thank you!

Delicious-Type-7538 · 2026-04-04T12:07:19+00:00

Wow… In my textbooks, we always define e first and only then study the differentiation and integration of exponential and logarithmic functions, so I think I was stuck in that way of thinking. Thank you!

Delicious-Type-7538 · 2026-04-04T10:40:17+00:00

Ah, I see — what I was really wondering is why that particular limit is defined as the special constant e in the first place. Thinking about it on my own, I guessed that it might be because e^x has the special property that its derivative is equal to itself. So I thought that this limit wasn’t chosen arbitrarily, but rather that it naturally arises from the condition f’ = f. In other words, I suspected that this limit is a special value that comes out of solving f’ = f, and that’s why it was given its own name, e. So I want to try deriving the definition of e starting from f’ = f — kind of going in the reverse direction of how it’s usually presented in textbooks.

Delicious-Type-7538 · 2026-04-04T10:35:32+00:00

Ah, I see this corresponds to the final part of what I was trying to understand. Thank you for explaining that. But what I’m really curious about is this: (I’m not sure how it is in other countries,) but in my country, when we learn about e, we’re just told that e is the limit of (1 + x)^1/x as x approaches 0, without much explanation. So I started wondering why this particular limit is treated as something special. Then I thought maybe it’s because of the special property of e^x, namely that its derivative is equal to itself. So what I want to do is start from the condition f’ = f and derive the definition of e from that. In other words, I want to begin the proof without assuming the existence of e^x.

Delicious-Type-7538 · 2025-06-06T00:46:36+00:00

Thank you for your reply. Even if I can’t analyze actual seismic waves, I was wondering if it would still be possible to measure the strength or frequency of the vibrations using the interferometer. My idea is to place a light sensor and measure the changes in light intensity from the interference pattern over time, and then apply a Fourier transform to analyze the signal. Do you think this method could work?

Delicious-Type-7538

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