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Can you construct a function $f$ which has produces a unique derivative at $f'$, $f''(x)$, etc, so that you could infinitely take derivatives and always have a never before seen function?

This would exclude cyclical derivative functions like $\sin(x)$ (non-unique derivatives) and functions that eventually arrive at a constant, as they would have the derivative 0 twice.. Would there be any functions like this?

Ben G
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    How about $e^{2x}$ – Kavi Rama Murthy Jan 11 '22 at 23:41
  • Works for me, curious if anyone would be willing to elaborate on what constraints there are, for example with $e^{2x}$ it varies only on coefficient.. is that a constraint? I'm guessing not. At this point I feel the example answers the question but curious about anything interesting math that may exist around this. Thanks Kavi! – Ben G Jan 11 '22 at 23:44
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    What about $f(x)=x e^x$? Then $f^{(k)}(x) =(x+k)e^x$ which is different for each $k$. – JackT Jan 11 '22 at 23:45
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    Does it have to be an entire function or is something like $f(x)=\frac1x$ acceptable? – bof Jan 11 '22 at 23:50
  • $x^x$ or anything involving nested functions will quickly get nasty. However unlike integration, taking derivatives of elementary functions will always return an elementary function, so you cannot go too far into exotic land by taking derivatives of elementary functions. – Tob Ernack Jan 11 '22 at 23:53
  • @bgcode See Functions that are their Own nth Derivatives for Real $n$ then choose any non-polynomial function which is not among those. – dxiv Jan 12 '22 at 00:04

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There are bunch of examples(such as $e^{ax}, xe^x$), but I think Hermite polynomial can be the best example.

Hermite polynomial are mutually orthogonal($\int_{-\infty}^\infty H_m(t)H_n(t)e^{-\frac{t^2}2}dt=\sqrt{2\pi}n!\delta_{mn}$) and forms an orthogonal basis of the Hilbert space.

The defenition of Hermite polynomial is $H_n(t)=(-1)^ne^\frac{t^2}{2}\frac{d^n}{dt^n}e^{-\frac{t^2}{2}}=\exp(-\frac12\frac{d^2}{dt^2})t^n$.

So $e^{-\frac{t^2}{2}}$ could be an answer. Its n-th derivative is $(-1)^ne^{-\frac{t^2}2}H_n(t)$.