The algebra does seem more transparent when you use an infix lerp operator: a [t] b meaning (1-t) a + t b. I also tweaked the notation further with (a..b) to me

The algebra does seem more transparent when you use an infix lerp operator: a [t] b meaning (1-t) a + t b. I also tweaked the notation further with (a..b) to mean (b-a) but thought of as "the vector from a to b". Then, for example, a quadratic spline: d/dt ((a [t] b) [t] (b [t] c)) = 2 (a..b) [t] (b..c) (there are some convenient laws to keep the intermediate steps visualizable.) From this it's clear that the tangents to the start and the end of the curve (at a and c) intersect at b: at t=0 it goes through a with derivative 2(a..b), and at t=1 it goes through c with derivative 2(b..c). Compare https://en.wikipedia.org/wiki/B%C3%A9zier_curve#Quadratic_B.C3.A9zier_curves https://en.wikipedia.org/wiki/B%C3%A9zier_curve#Quadratic_B.... in the usual notation.