Backpropagation: The Step-by-Step Derivation

Backpropagation applies the chain rule to share a network's error layer by layer. You first compute the error of the output layer, propagate it backward, and use it to obtain the gradients of every weight and bias in a single pass, summarised in four compact equations you can code directly.

July 16, 2026 8 min

Technology

The Chain Rule, the Engine of Backpropagation

The chain rule computes the derivative of a composite function by multiplying the derivatives of its links: if y depends on u and u depends on x, then dy/dx equals dy/du times du/dx. That layer-by-layer multiplication of derivatives is exactly what backpropagation does to train a neural network.

July 16, 2026 6 min