I asked this at Math.StackExchange without any success (link):

What's the best way (or does it even make sense) to measure the instantaneous rate of change in the median of a collection of functions (where the median function at is defined at some $x$ as the point-wise median of the functional values at $x$)?

Set-up: I have a finite collection of continuous, differentiable real functions on an interval, $\{f_i(x)\}$. Since they are not nicely distributed (e.g. on a small subinterval some $f_j$ might be an extreme outlier), the point-wise median, $m(x) = \text{median}(\{f_i(x)\})$, is a better measure of the center than the average.

Question: How do you measure/compute/define the instantaneous rate of change in $m$?

Really, for some small interval around a given $x$, $m$ is just one of the $f_i$. But although this $f_i$ is in the center, how it's changing might be vastly different then how the bulk of functions are progressing. For instance, in general the collection of functions could be increasing (as evidenced in a plot or by general understanding of the contextual problem), but in the moment that $f_i(x)$ is central, $f'_i(x)$ could be negative. So does it make (more? any?) sense to define $m'(x) = \text{median}(\{f'_i(x)\})$?

Here's a similar but different situation where the answer is obvious: If you use the average instead of the median, then, since the average is just a linear combination of the functions, the derivative of the average is the average of the derivatives. So what's the analogous answer for the median? Also, whatever works for the median should also probably work for any percentile function since the median is just the 50-percentile.