Each weight gradient is the upstream gradient times the input feeding that weight. This lesson names those derivatives parameter by parameter, so each exact fraction is tied to the weight it describes. The branch with zero upstream gradient contributes exact zeroes, not approximate small values.

highlighted = computed this step

First hidden-unit gradients

For the first hidden unit, dL/dz1=-2. Multiplying by x1 and x2 gives dL/dw11=-2 and dL/dw12=-4. Each one is the exact derivative for that named weight: upstream pre-activation gradient times the local input.

dLdw11=21=2,dLdw12=22=4\frac{dL}{dw_{11}}=-2\cdot1=-2,\quad \frac{dL}{dw_{12}}=-2\cdot2=-4
Weight gradientsParameter gradients are recomputed by the chain rule.Weight gradientsParameter gradients are recomputed by the chain rule.one reverse-chain step (all exact)quantityrulevaluedL/dyhat2*(2-3)-2dL/dv1-2*2-4dL/dv2-2*00dL/dcdL/dyhat-2dL/dh1-2*1-2dL/dh2-2*1-2ReLU'(z1)z1=2>01ReLU'(z2)z2=-1<=00dL/dz1-2*1-2dL/dz2-2*00dL/dw11-2*1-2dL/dw12-2*2-4dL/db1dL/dz1-2dL/dw210*10dL/dw220*20dL/db2dL/dz20

Biases and the blocked unit

The first bias gradient is dL/db1=-2. Since dL/dz2=0, the second hidden unit has zero weight and bias gradients. A closed ReLU gate upstream makes every local multiplication on the second branch land at zero.

dLdb1=2,dLdw21=dLdw22=dLdb2=0\frac{dL}{db_1}=-2,\quad \frac{dL}{dw_{21}}=\frac{dL}{dw_{22}}=\frac{dL}{db_2}=0
Weight gradientsParameter gradients are recomputed by the chain rule.Weight gradientsParameter gradients are recomputed by the chain rule.one reverse-chain step (all exact)quantityrulevaluedL/dyhat2*(2-3)-2dL/dv1-2*2-4dL/dv2-2*00dL/dcdL/dyhat-2dL/dh1-2*1-2dL/dh2-2*1-2ReLU'(z1)z1=2>01ReLU'(z2)z2=-1<=00dL/dz1-2*1-2dL/dz2-2*00dL/dw11-2*1-2dL/dw12-2*2-4dL/db1dL/dz1-2dL/dw210*10dL/dw220*20dL/db2dL/dz20

Summary

The parameter-gradient register is exact: dw11=-2, dw12=-4, db1=-2, and the dead branch entries are all 0. The table is the chain rule applied edge by edge to the exact forward graph.

dw11=2,dw12=4,db1=2dw_{11}=-2,\quad dw_{12}=-4,\quad db_1=-2
Weight gradientsParameter gradients are recomputed by the chain rule.Weight gradientsParameter gradients are recomputed by the chain rule.one reverse-chain step (all exact)quantityrulevaluedL/dyhat2*(2-3)-2dL/dv1-2*2-4dL/dv2-2*00dL/dcdL/dyhat-2dL/dh1-2*1-2dL/dh2-2*1-2ReLU'(z1)z1=2>01ReLU'(z2)z2=-1<=00dL/dz1-2*1-2dL/dz2-2*00dL/dw11-2*1-2dL/dw12-2*2-4dL/db1dL/dz1-2dL/dw210*10dL/dw220*20dL/db2dL/dz20