Time evolution on the trajectories of a dynamical system

Information about the time evolution of a certain function, for example f[x(t), y(f)] when x(t), y(t) are trajectories of a dynamical system, i.e. satisfying equations (Al), is frequently useful. The derivative of a function / with respect to time may be readily calculated provided that x, y evolve with time in accordance with equations (Al). Such a derivative is called the derivative of a function / (with respect to time) on the trajectories of a dynamical system (Al) or the (time) derivative with respect to the system (Al) and denoted f 

Using the formula for the derivative of a superposition of functions, followed by an application of equations (Al) 

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