Let's generalize a bit to say we want to find the quickest path across two media:
We are traveling from point A to point B then to point C by the quickest path possible. The question is, where should point B be placed?
I have set up xy coordinate axes such that:
The distance from A to B is:
The distance from B to C is:
The speed through medium 1 is and the speed through medium 2 is . We assume both velocities are positive values. Thus, using the relationship between distance, constant velocity and time, we find the total time as a function of B is:
Differentiating with respect to B, we find:
The denominator will always be positive, so we need only consider:
Let's take a moment to rewrite this:
Thus, we have shown that Snell's law (or Descartes' law) is satisfied when .