I had the same reaction. It looks like they fit a spline to the data, and then took the analytical derivative of the spline. But there are other ways to do it l

I had the same reaction. It looks like they fit a spline to the data, and then took the analytical derivative of the spline. But there are other ways to do it like locally weighted regression. In general estimating derivatives from real world data is noisy, ill conditioned, should be treated carefully.