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I was wondering how you could find all functions $f$ for which it holds true that $f^{-1} = f'$.

I found the solution $\varphi^{1-\varphi}x^{\varphi}$ for which this holds true. Are there any other solutions and is there a closed form for all solutions for this equation?

Thanks in advance.

IV_
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  • Check this: https://math.stackexchange.com/q/3312572/42969 – Martin R Aug 22 '22 at 15:57
  • see "2.) Non-Existence of Non-elementary Integrals" in my answer at https://math.stackexchange.com/questions/3095196/functions-that-are-easy-to-antidifferentiate-but-whose-inverses-are-hard-to-anti/3095634#3095634 – IV_ Aug 22 '22 at 16:15
  • Your class is not correct, since for $\phi\equiv,1$ we get $f(x)=x$ which does not satisfy the condition, because $f'(x)=1$ and $f^{-1}(x)=x$ –  Aug 22 '22 at 17:39
  • @Oxff: Trying to make sense of this An inverse function is a points set. The derivative is a single value found at a given point in the function set. – Narasimham Aug 22 '22 at 18:49
  • @GeorgeTsoutsinos, Phi here denotes the golden ratio, not an arbitrary parameter, and the given example does work. – Ned Aug 22 '22 at 21:10
  • ΟΚ you should have clarified that! –  Aug 22 '22 at 21:33

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