Backpropagation Explained for Beginners (Part 3): How Backpropagation Really Works

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This is the third installment in an educational series on backpropagation for ML beginners, published on Towards Data Science. The central insight of this part: when computing gradients for all parameters in a neural network, many intermediate partial derivatives appear in multiple equations. Recomputing them from scratch for each parameter is wasteful; the chain rule naturally enables gradient reuse, which is why backpropagation is efficient at scale.\n\nThe article demonstrates this concretely with a small network containing weights w1–w4 and biases b1–b3. The partial derivative of loss with respect to the output ŷ appears in every parameter's gradient equation. Storing and reusing this intermediate value, rather than recomputing it, is the core computational efficiency of backpropagation.\n\nThis is educational writing for practitioners learning fundamentals, not research or product news. Author Nikhil Dasari builds directly on Parts 1 and 2 of the series. The mathematical exposition is clear and accessible, bridging theory and practical understanding of why modern deep learning frameworks are efficient.