Backpropagation computes gradients layer by layer from the output back to the input, multiplying local derivatives at each layer using the chain rule - efficiently propagating the loss gradient to every weight.

Forward pass: compute activations at each layer, storing intermediate values. Backward pass: start from the loss ∂L/∂ŷ, then apply chain rule: ∂L/∂wₗ = (∂L/∂aₗ₊₁) × (∂aₗ₊₁/∂wₗ). Each layer multiplies the incoming gradient by its local Jacobian. Vanishing gradients occur when layer derivatives are small (sigmoid activations < 0.25) - gradients shrink toward zero in early layers, slowing learning. ReLU mitigates this because its derivative is either 0 or 1.