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Let $d(x,y)$ be a distance metric for two points $x,y\in \mathbb{R}^p$. Further, suppose that there are two real or complex sequences $X_n(x)$ and $X_n(y)$, $n=1,2,\ldots$ that depend on $x$ and $y$ respectively. Let $D(X_n(x),X_n(y))$ be a distance metric between two such sequences. Then one can form some kind of derivative by taking the limit $\delta \rightarrow 0$ of

$${ d(x,x+\delta)} \over {D(X_n(x),X_n(x+\delta)) }$$

The question is simply, does this kind of derivative have a name? Are there some particular distance measures that are frequently used in this context? I would appreciate any pointers to further reading.

A further question on notation, since this appears to have caused some confusion: What would be a good way of indicating that an infinite sequence $X_n$ depends on the parameters $a \in \mathbb{R}^p$? Is there any other notation that is prefered over the one I used, $X_n(a)$?

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I don't understand the notation in your question (is $D$ somehow related to $d$, for one thing) but I can suggest this article: metric differential –  user53153 Jan 27 '13 at 16:05
    
Thanks for the link. So, I might call this idea a metric differential then. –  Oberdada Jan 27 '13 at 20:59
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