What are the relationships between Functional Reactive Programming, Automatic Differentiation, Self-Adjusting Computation and Partial Evaluation?
I essentially agree with Dave's comment: these are lots of different things (and I only feel remotely competent to talk about three of them). 

However, from the perspective of a consumer of functional programming research, the three things I know something about probably feel related, because they all are ways of making functional programs deal with inputs that change over time. For instance:


Functional reactive programming - consider this when you want to write a program as if you're taking, as input, list of all the user events that's ever happened, but you still want the program to behave reasonably as you feed it inputs over time. An underlying assumption is therefore that the data about past events won't change, and that the only changes to the input will be the addition of new events that happened after all the old events. 
Partial evaluation - consider this when your program neatly cleaves into two parts, and you want to recompute, over and over, part 2 based on the outcome of part 1. This assumes your code has the property that the difference between "part 1" and "part 2" are known when you write the code, so an underlying assumption is that the data that stays the same (the data part 1 computes with) and the data that changes (the data part 2 computes with) are known in advance.
Self-adjusting computation - consider this when your program's execution flow is not neatly divided into parts, but involves a lot of decisions based on data. Self-adjusting computation is designed to react to small changes in inputs by only redoing computation that was based on modified data. Therefore, an underlying assumption is that the particular changes can't be anticipated in advance.


[Update] In practice, the challenges associated with each of these three things are different: it doesn't necessarily make sense to talk about any of them as special cases of the others. However, as Jake points out below, there are (slightly restricted) libraries of functional reactive programming that are implemented on top of self-adjusting computation primitives. Similarly, it might be possible to effectively implement something like partial evaluation on top of self-adjusting computation, since SAC makes the very few a priori assumptions about its input data.