This paper provides some insight into the gradient based training of adaptive dynamic systems such as recurrent neural networks or neural network based controllers. In the neural network literature, training algorithms for such systems are generally of two types: those which propagate derivative information forwards in time, and those which propagate it backwards. These two types of algorithm are derived and analyzed for a simple prototype system. It is shown that they are very closely related because they compute the same components of the gradient vector but in a different order. The well known computational properties of each algorithm are then explained using a simple matrix multiplication analogy. Extensions of the prototype to control systems are demonstrated.