I have found that the function
$$f(x) = \left(\frac{1}{2} \left(\sqrt{5}+1\right)\right)^{\frac{1-\sqrt{5}}{2}} x^{\frac{\sqrt{5}+1}{2} }$$
satisfies the relation $$f^{-1}(x)=f'(x)$$
Is this solution unique? If so, how to prove it?
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Raffaele
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1This has been asked and answered before: https://approach0.xyz/search/?q=%24f%5E%7B-1%7D(x)%3Df%27(x)%24 – Martin R Oct 19 '20 at 13:30
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1https://math.stackexchange.com/q/279332, https://math.stackexchange.com/q/735695. https://math.stackexchange.com/q/279517, https://math.stackexchange.com/q/1916191 – Martin R Oct 19 '20 at 13:31
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@MartinR Thank you! I searched MSE but couldn't find anything – Raffaele Oct 19 '20 at 13:46
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https://www.youtube.com/watch?v=rNUfiQgj6ZI&list=LL&index=124&t=1s – Physor Oct 19 '20 at 14:02
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Also with Dr Peyam: https://www.youtube.com/watch?v=0IlWyIaMXqI – md2perpe Oct 19 '20 at 14:10