The exchange rate is not exactly a continuous-time signal. It is updated at discrete time instants and, therefore, it seems more natural to model it as a discrete-time signal, imho. What exactly does "noisy" even mean in this context? I assume you meant to say that the signal contains significant high-frequency content.

The first thing you might want to do is to use interpolation to obtain a discrete-time signal that could be thought of as the uniform sampling of a continuous-time signal. Then, you can use a smoothing filter (e.g., moving average) to remove the high-frequency content. Beware that causal smoothing filters introduce a delay. Then, once the high-frequency content has been removed, you can use a discrete-time differentiator to obtain an estimate of the derivative of the continuous-time signal that we imagine to be the originator of the time-series.

Books you might want to take a look at:

The Scientist and Engineer's Guide to Digital Signal Processing

A First Course on Time Series Analysis