The time derivative of the position is simply the velocity, rotated
into the coordinate system of the position variables. For simulators
that store the aircraft's CG position in local coordinates, the
kinematic equations for position are:

= | C_{11}u + C_{12}v + C_{13}w |
(33) | |

= | C_{21}u + C_{22}v + C_{23}w |
(34) | |

= | C_{31}u + C_{32}v + C_{33}w |
(35) |

Simulators that store position in a Cartesian coordinate system other than local coordinates substitute components of the appropriate rotation matrix into Equations 33-35.

Simulators that store aircraft position in geocentric spherical
coordinates can calculate the spherical coordinate time derivatives
using the using the local coordinate time derivatives as intermediate
variables
: