The chain rule is the rule that backpropagation is built on. It tells you how to differentiate a composition: a function inside another function, like f(g(x)).
To differentiate "outer of inner," take the outer derivative (leaving the inner alone), then multiply by the inner derivative. The rates of change multiply along the chain.
Think of it as a pipeline: x → g → f. A nudge in x gets amplified by g′, then that nudge gets amplified again by f′. The total amplification is the product of the two. The figure traces the derivatives multiplying along the composition.
Where this lives in MLBackpropagation is the chain rule, run backwards through a network. A deep net is one giant composition (layer after layer after layer), and the gradient of the loss with respect to an early weight is a product of local derivatives, one per layer, multiplied along the path. This is why "vanishing gradients" happen: multiply many small derivatives and the product shrinks to nothing. The chain rule…