Examining e^x, a curious function that is its own derivative.

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What’s so special about e^x?

All right, so f’(x)=f(x), so what?

This is interesting. It appears as though exponential functions are the only ones of the form dR/dt=kR

That is to say, the growth at any given time is determined by the current value at a time. In the case of e^x, k would be equal to 1.

We should investigate other interesting cases; what about something that fits the form 2f’(x)=f(x)? And what of xf’(x)=f(x)? These are just simple differential equations that I don’t really have a great grasp on.