In the course of my work, I found a problem where the ratio between the 3rd and the 1st derivatives of a function is used as some kind of measure of how smooth the function is. The derivatives are computed numerically. If the ratio is large, the function is not smooth enough.

I made several examples to explore this idea, but the point still escapes me. If we take the first derivative as a measure how much function locally deviates from a constant and the third derivative as a measure how much it deviates from a parabola, then the ratio of the two will be large when it does deviate from parabola, but also when the deviation from a constant it small. It seems like a rather ill-defined quantity.

Does this ratio appear in some formal mathematical theory in this context?