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Backpropagation is a fundamental algorithm used for training artificial neural networks. It is an implementation of the gradient descent method that allows the network to adjust its weights to minimize the error between the predicted output and the actual target values. The essence of backpropagation lies in calculating the gradient of the loss function with respect to each weight by applying the chain rule of calculus, propagating the error backward through the network. In a typical feedforward neural network, there are layers of neurons including an input layer, one or more hidden layers, and an output layer. Each neuron receives inputs, applies a weight to each input, sums them up, and passes the result through an activation function to produce an output. During training, the network makes predictions for a given input, compares them to the expected outputs, calculates the error, and updates the weights to minimize this error iteratively. To illustrate the backpropagation algorithm, let us consider a simple neural network with two input neurons, one hidden layer with two neurons, and one output neuron. Assume the network uses a sigmoid activation function, which is defined as σ(x) ______ ___ ______ _______ ______ ______.
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