Is diffraction empirically derived or based off something else?

Diffraction can be derived from the wave equation and basic principles, but the empirical observations came first, historically. So, as stated, the answer is: "both"

Historically, diffraction was observed before it was explained with the wave equation. Young's double slit experiment started the idea of light as a wave more than 50 years before Maxwell's equations (the "base principles" behind diffraction).

The wave equation also predicts diffraction. The math can be somewhat involved, but it can be summarized as follows: In 3 dimensions, only a plane wave can travel in just one direction - if the wave has any limits (i.e. if it has any cross-sectional shape), those limits (its shape), it must change as it travels. If you fix the wave shape at one point in the direction of travel (say, by using the shape of Young's double slit), then you can very accurately calculate a diffraction pattern somewhere further along the direction of travel.

There are many good resources which explain diffraction, including https://en.wikipedia.org/wiki/Diffraction and any good optics textbook (unless it only has geometric optics).