Compose two operations
Let u = 2w and L = u². A change in w affects u, which affects L. The derivative is dL/dw = (dL/du)(du/dw) = 2u × 2 = 4u = 8w.
Check one number
At w = 3, u = 6 and L = 36. The derivatives are dL/du = 12 and du/dw = 2, so dL/dw = 24. A small increase of 0.001 in w predicts an increase of about 0.024 in L; the exact increase also contains a tiny higher-order term. Follow w through the intermediate value u to the loss. The backward derivative is listed beneath the diagram.
INTERACTIVE EXPLORATION
Trace a change through two operations
Change w and compare each local sensitivity. The overall derivative is their product.
Multiple paths
When a value affects an output through several paths, each path contributes and the contributions are added. Backpropagation applies this local bookkeeping efficiently across a computation graph.
Check your understanding
At w = 3, what is dL/dw for L = (2w)²?
24. Multiply the local sensitivities 12 and 2.