  Let $T$ be a complete, model complete o-minimal theory extending the theory
of real closed ordered fields and assume that $T$ is power bounded. Let $K$ be
a model of $T$ equipped with a $T$-convex valuation ring $\mathcal{O}$ and a
$T$-derivation $\partial$ such that $\partial$ is monotone, i.e., weakly
contractive with respect to the valuation induced by $\mathcal{O}$. We show
that the theory of monotone $T$-convex $T$-differential fields, i.e., the
common theory of such $K$, has a model completion, which is complete and
distal. Among the axioms of this model completion, we isolate an analogue of
henselianity that we call $T^{\partial}$-henselianity. We establish an
Ax--Kochen/Ershov theorem and further results for monotone $T$-convex
$T$-differential fields that are $T^{\partial}$-henselian.
