Its easy to visualize what's going on with the derivatives of x**2 and x**3 with the usual square and cube representations of those functions: a square can be enlarged by "building out" along two edges, a cube can be likewise by "building out" on three faces -- the "error" artifacts are the little dx corner square in the 2D case, the corner cube as well as the three edge "lines" in the 3D case. Its just a cute way of seeing where the derivatives d/dx(x**2) = 2x, and d/dx(x**3) = 3(x**2) come from.
But I don't see how to do anything similar with triangles or tetrahedrons. Perhaps Kirby will show us. Or does this simple exercise point to something a bit more fundamental than simple "cultural choice"?