The three types of derivatives: vectors-by-matrices, matrices-by-vectors, and matrices-by-matrices. These are not as widely considered and a notation is not widely agreed upon. However the second type (matrices-by-vectors) is widely used in Robotics.

If we consider the following state form for a robot:

The matrix is skew-symmetric.

Here the matrix-by-vector derivative is:

Note that this is not true if the robot dynamics is expressed in any other forms (for example).