$T$-convex $T$-differential fields and their immediate extensions

Let $T$ be a polynomially bounded o-minimal theory extending the of real closed ordered fields. $K$ model equipped with $T$-convex valuation ring and $T$-derivation. If this derivation is continuous respect to topology, then we call $T$-differential field. We show that every field has an immediate strict extension which spherically complete. In some important cases, assumption polynomial boundedness can relaxed power boundedness.