Derivation: Backward Functions for Basic Modules and Vectorized Backpropagation

The previous lesson stated the backpropagation algorithm and its chain-rule justification at the level of a single hand-worked example. This note derives, once and for all, the backward function for each of the three building blocks used throughout this topic — the affine (dense) layer, the ReLU activation, and the combined softmax-plus-cross-entropy output layer — and assembles them into a fully vectorised backward pass for an arbitrary-depth MLP.