I want to prove the following result: $$ {f^{-1}}'(x)=\frac{1}{f'({f^{-1}}(x))}$$ Is simple application of chain rule a valid proof of it?

Here is an alternative: You can use $$ f^{-1}(x)=\int\frac{1}{f'(f^{-1}(x))}\,dx + c. \tag{1} $$ from "Inverse functions and differentiation".

$\begingroup$ Yes .. if you have two identical functions they have the same derivation. I used this fact twice (for the first implication and for the second implication also) $\endgroup$ – …

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