The paper is not very specific regarding methodology for taking the derivative Sharpe ratio (DSR). Based purely on subtext, I am inferring that the author intends to differentiate the Sharpe ratio with respect to itself.
The closest that the author comes to specifying DSR:
Economically speaking, the derivative sharp ratio is
analogous to the marginal utility in terms of willingness to
bear how much risk for one unit increment of sharp ratio.
which makes me think that it might be the change in SR with respect to itself.
In any case, we begin with the definition of a derivative:
$$\frac{dS}{d\tau}=\lim_{\tau \to 0}\frac{S[t+\tau]-S[t]}{\tau}$$
The paper further says that rather than find a symbolic solution to the derivative, the gradient would be measured across multiple time steps. Thus, we can discretize $\frac{dS}{S}$ as such:
$$ \frac{\sum_{t}^T (S[\tau]-S[\tau-1])}{\sum_{t-1}^TS[\tau]}$$
...which expands to:
$$ \left(\frac{r_{a(T))}-r_{m(T)}}{\sigma_{a(T)}}-\frac{r_{a(t)}-r_{m(t)}}{\sigma_{a(t)}}\right) * \left(\frac{\bar{r}_a-\bar{r}_m}{\bar{\sigma}_a}\right)^{-1}$$
which gives us something akin to the author's specification for a the marginal utility function of an incremental gain in the SR. In this case, the "derivative Sharpe ratio" (DSR) is not really a performance metric itself, but rather a sensitivity metric to changes in risk adjusted performance.