The curve is not a cubic function AFAIK. It is more that the spline is used to fit, or approximate, the functions of photomaterials which include toe, mid scale and shoulder.
The first derivative is, of course the slope, and when you multiply the instantaneous slope of a film curve by the instantaneous slope of the paper curve gotten at a given exposure, then you have the instantaneous slope of the print you obtain.
This is only approximately true for B&W materials, but deviates in color unless you read analytical rather than analog densities. Then you need to add in to the mix, a 3x3 matrix of the dyes.
You can easily see how complex this becomes and why we used so much computation power in our work. Probably more than you wanted to know. But, I hope it helps.