Given a time series $\{V_t\}_{t=1}^{n}$, where $t \mapsto V_t \in \mathbb{R}$, I want to have some smoothed notion of the "derivative" of this time series. It was recommended that I look at
$$\tilde{V}_t = \frac{V_t - \text{$k$-step moving average of } V_t}{\text{$k$-step std dev of $V_t$}}.$$
The $k$-step moving average of $V_t$ is $\frac{1}{k} \sum_{i=0}^{k-1}V_{t-i}$.
This quantity, empirically, seems to act something like derivative. (For a line it is constant, for a sine curve it is almost a cosine curve, etc.) Can someone explain why, mathematically, this would be a heuristic for a derivative?
Thanks.