Quick reference
Two-layer backpropagation
For h = w₁x, o = w₂h, and L = (o − y)²:
dL/do = 2(o − y)
dL/dw₂ = (dL/do) × h
dL/dh = (dL/do) × w₂
dL/dw₁ = (dL/dh) × x
wᵢ ← wᵢ − η(dL/dwᵢ)
The output weight receives the error signal directly. The first-layer weight receives it through the output layer, which is why its gradient includes w₂. This is “credit assignment”: how much each earlier component contributed to the final error.
Primary source: Rumelhart, Hinton & Williams (1986).