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Let $k$ be a finite field and $k_E:=k((X))$ denote the field of Laurent series over $k$. We define a Frobenius endomorphism on $k_E$ via $f(X)\mapsto f(X^p)$. We choose a lift $\varphi:k_E^{sep}\...

Before posting I want to make it clear that I posted the same question in stack exchange awhile ago (https://math.stackexchange.com/questions/1533814/property-of-derivative-in-a-local-field) but didn'...