What is solution of $f'(f(x))=\exp(f'^{-1}(x))$ with $ f'^{-1}$ is a compositional inverse of $f'$?

Question

,Assume $f$ a bijective and differentiable function on its domain , I want to find solutions of this functional such that $f\colon\mathbb{R} \to \mathbb{R}$; $f'(f(x))=\exp(f'^{-1}(x))$ such that $f'^{-1}$ is a compositional inverse of $f'$ , I have tried $f(x)=-x$ seems works because we have $f'(f(x))$ is increasing function in the same time $f^{-1}$ exist and would be increasing imply $f'^{-1}$ exist and increasing , if we raise exponential we have increasing functions, but am not sure about this solution ? Is there a clear solution, because its seems that it has a nontrivial solution which it is a formal power series arround $x=0 $ , we may get its coefficients using inverse function theorem ?