Author(s): Kamrun Nahar

Originally published on Towards AI.

The chain rule is the most used rule in data science. Here is how it works, why it is true, and where you meet it again and again later.

The last time I changed money, I did calculus without noticing.

The article explains the chain rule through everyday “machines” and animated examples: when one function feeds another, you multiply the local slopes in sequence (e.g., chained currency exchange rates), and the final change depends on both the inner and outer stages. It highlights common student mistakes—especially dropping the inner derivative or reading the outer slope at the wrong value—showing how these errors can look correct on small tests but fail dramatically in real training. It then discusses why the usual “cancelling” intuition is misleading (division by Δu can fail when the inner change hits zero) and replaces it with a geometric/limit intuition: locally, differentiable functions behave like lines, so multiplying slopes is justified. The author extends the idea to more complex cases: when there are multiple paths, you multiply along each path and add across paths (mirroring multivariable chain rules and how backpropagation accumulates gradients). Finally, it connects the chain rule directly to statistics and machine learning—likelihoods, transformation density factors, backpropagation, vanishing gradients, and the practical “fixes” like ReLU—emphasizing that the rule is the core engine behind how models learn and how uncertainty propagates.

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Published via Towards AI