This is mostly just a note to myself, but in case anyone else is interested: Steve Walker has a nice post walking you through the derivation of the continuous time Price equation. The discrete time form is much more commonly seen (e.g., in my own work), but the continuous time form crops up in ecology, for instance in some very nice work by Steve Ellner.
Steve also points out that the continuous time Price equation is basically an instance of a mathematical rule called the chain rule. Which raises the deep and interesting question of the boundary between abstract mathematical rules and concrete mathematical models of specific real-world systems and phenomena. Where does math-as-abstraction or math-as-rules-of-logic stop and math-as-description-of-real-things begin? The line can be rather fuzzy, which is something I’ll probably be posting about more in the near future.