Alternative to Extended Kalman Filter When Prediction Function Is Not Differentiable
I will tell you something, even if it is differntiable, use Unscented Kalman Filter for any non linear case.

This flavor of Kalman Filter, based on the Unscented Transform, is almost always superior to the Extended Kalman due to its properties.
The main reason is it is able to better predict the mean and variance (Which all Kalman Filter needs) of the distribution after the non linear function applied.

In most cases it is even simpler to calculate hence more computationally efficient.
Think of it as Poor's man Particle Filter (Though the sampling points are deterministic).