I derive the important equation that relates the time derivative of a vector computed in one frame to that computed in another frame. I make the point that we must understand the distinction between frames and coordinates to appreciate what the equations are saying. That discussion leads naturally to the concept of centrifugal and Coriolis forces in rotating frames. I use the frame-dependent time derivative to derive some equations for robotics, and finish with a wider discussion of the time derivative for tensors and in fluid flow.