Backpropagation through time is a powerful method for calculating derivatives in dynamic systems, applicable to neural networks, econometric models, fuzzy logic, and fluid dynamics. It extends basic backpropagation to handle time-dependent data, enabling optimization of iterative procedures, neural networks with memory, and control systems. The method uses the chain rule for ordered derivatives, allowing efficient computation of gradients for large systems. The paper reviews basic backpropagation, introduces backpropagation through time, and discusses its applications in pattern recognition, systems identification, and control. It also covers extensions to handle recurrent networks, simultaneous equations, and other practical issues. Pseudocode is provided to illustrate the algorithms. The method is particularly useful for dynamic systems where past states influence future predictions, such as in speech recognition or control systems. The paper emphasizes the importance of careful implementation, noting that backpropagation through time requires backward propagation of derivatives through time, which can be computationally intensive. It also discusses alternative methods and extensions, including the use of non-neural networks and adaptive critic methods for handling noise and real-time learning. The paper concludes with applications in neuroidentification and neurocontrol, where backpropagation through time helps in modeling and controlling dynamic systems.Backpropagation through time is a powerful method for calculating derivatives in dynamic systems, applicable to neural networks, econometric models, fuzzy logic, and fluid dynamics. It extends basic backpropagation to handle time-dependent data, enabling optimization of iterative procedures, neural networks with memory, and control systems. The method uses the chain rule for ordered derivatives, allowing efficient computation of gradients for large systems. The paper reviews basic backpropagation, introduces backpropagation through time, and discusses its applications in pattern recognition, systems identification, and control. It also covers extensions to handle recurrent networks, simultaneous equations, and other practical issues. Pseudocode is provided to illustrate the algorithms. The method is particularly useful for dynamic systems where past states influence future predictions, such as in speech recognition or control systems. The paper emphasizes the importance of careful implementation, noting that backpropagation through time requires backward propagation of derivatives through time, which can be computationally intensive. It also discusses alternative methods and extensions, including the use of non-neural networks and adaptive critic methods for handling noise and real-time learning. The paper concludes with applications in neuroidentification and neurocontrol, where backpropagation through time helps in modeling and controlling dynamic systems.