It is widely known that every derivaitve can be represented as partial case of derivative on time scales, like for any function $f\colon \mathbb{T} \to \mathbb{R}$ and $t\in\mathbb{T}^{\kappa}$ then $f^{\Delta}(t)$ is time-scale derivative of $f$: \begin{equation*} f^{\Delta} (t) = \frac{f(\sigma(t) - f(t)}{\mu(t)}, \end{equation*} where $\sigma(t)$ is forward jump operator and $\mu(t) = \sigma(t) - t$.
However, it is unclear (at least for me) which notation to use when it is necessary to express some particular derivative in terms of time scales. E.g when $\mathbb{T} = \mathbb{R}$ we have usual derivaitve $f'(x)$ etc. Still, we have a Jackson's $q$-derivative which is defined as
\begin{equation*} D_{q} f(x) = \frac{f(qx)-f(x)}{qx-x}, \quad x\neq 0 \end{equation*}
Which is correct notation to express, for example Jackson derivative in terms of time scales? How to denote time scale defined by function $u(t)$ for example?
My approach:
For every $x\neq 0, \; x\in \mathbb{T} = \{tz\colon t\in\mathbb{R}, t=const, z\in \mathbb{R} \}$
$$D_qf(x) = f^{\Delta}(x)$$