This question is basically a special case of this older question of mine, which is still unanswered.
Let $\mathcal{D}$ be the field of d.c.e. reals; these turn out to be exactly the reals $\alpha$ for which there is a computable sequence of rationals $(\alpha_s)_{s\in\mathbb{N}}$ such that $\lim_{s\rightarrow\infty}\alpha_s=\alpha$ and $\sum_{i\in\mathbb{N}}\vert\alpha_i-\alpha_{i+1}\vert<\infty$ (call such a sequence a "d.c.e. approximation"). It turns out that $\mathcal{D}$ has a natural derivation, namely $$\partial \alpha:=\lim_{s\rightarrow\infty}{\alpha-\alpha_s\over \Omega-\Omega_s}$$ where $(\alpha_s)_{s\in\mathbb{N}}$ is a d.c.e. approximation of $\alpha$, $\Omega$ is Chaitin's constant, and $(\Omega_s)_{s\in\mathbb{N}}$ is a d.c.e. approximation of $\Omega$. The constant subfield of the differential field $(\mathcal{D},\partial)$ is the field $\mathcal{N}$ of non-random d.c.e. reals. See these slides of J. Miller for more on the above.
My question is whether there is a way to construe elements of $\mathcal{D}$ as functions on non-random d.c.e. reals such that the chain rule holds, as in the above-linked question. In the language of that question, I'm asking whether $(\mathcal{D},\partial)$ is concrete; the full definition is a bit messy, but basically I want to know if $(\mathcal{D},\partial)$ can be identified with some sub-differential algebra of a differential algebra consisting of partial functions on $\mathcal{N}$ which is closed under composition and in which an appropriate version (i.e. allowing small amounts of partiality) of $\partial (f\circ g)=((\partial f)\circ g)\cdot \partial g$ holds.
Since $\mathcal{D}$ is countable the answer is almost certainly "yes" via an unnatural construction; I'm particularly interested in whether there is a "meaningful" way of construing d.c.e. reals as functions on the nonrandom d.c.e. reals, although I don't know how to make this precise.
