Let $a$ be algebraic over $K:=\mathbb F_q\left(\left(\frac1T\right)\right)$. Can one extend continuously the hyperderivatives on $K(a)$?

Recal that the hyprderivative $D_h$ over $K$ is defined by
$$D_h(\sum_{i\le m}\frac{a_i}{T^i})=\sum_{i\le m}\binom{-i}h\frac{a_i}{T^{i+h}}$$
with $\binom{-i}h=\frac{-i(-i-1)\cdots(-i+h-1)}{h!}$