Let $q$ be a power of a prime $p$ and $w$ be an irreducible polynomial of $\mathbb F_q[T]$. Denote by $\mathbb C_w$ the completion of an algebraic closure of $K_w$, the completion of $\mathbb F_q(T)$ for the normalized valuation $v$ ($v(w)=1$). Denote by $K^{\text{sep}}_w$ the separable closure of $K_w$ in $\mathbb C_w$. One knows that the derivative on $\mathbb F_q(T)$ can be extended uniquely in $K^{\text{sep}}_w$ (one still denotes it by $'$). My question: for every $\alpha\in K^{\text{sep}}_w$, does one have $v(\alpha')=v(\alpha)-1$ or $v(\alpha')\ge v(\alpha)$ if $p\mid v(\alpha)$. Obviously, it is true for $\alpha$ in the completion of $\mathbb F_q(T)$.