So I know one (obvious) way to differentiate the cubic bezier curve point function P(t) = GBT:
expand the terms (B * T), differentiate them, collect the coefficents and then form the coefficent matrix (spline matrix) B again but I'm wondering if there's and easier way to go about this (namely, how to find B' without having to expand the terms GT out every time I want to differentiate and find P'(t). Essentially I want a way to find B' in the easiest (and fastest) way and I'm wondering if a method exists for something like this.
$$P(t)=\pmatrix{P1 & P2 & P3 & P4} * \pmatrix{ 1 & -3 & 3 & -1 \\ 0 & 3 & -6 & 3 \\ 0 & 0 & 3 &-3 \\ 0 & 0 & 0 & 1} * \pmatrix{1\\t\\t^2\\t^3}$$
You can just differentiate the T so that it becomes $T'(t)=<0, 1, 2t, 3t^2>$ and $P'(t)=GBT'(t)$