Calculus I · 2A · exploration
The Chain Rule as Composition of Local Linear Maps
Explore the chain rule as composition of local linear maps in an optional advanced note connecting Calculus I to later analysis and mathematical theory.
Section overview
Optional advanced explorationsWhat this section is building
Explore the chain rule as composition of local linear maps in an optional advanced note connecting Calculus I to later analysis and mathematical theory.
Differentiability is a local linear approximation property with consequences beyond computation.
Return here after the core path is secure, and connect each abstraction to a concrete derivative example.
Collecting formal language without linking it to the local linear model it describes.
Why the Derivative Factors Multiply
Write local models
and
Substitute the change in for . To first order,
The coefficient of is the chain-rule product.
Read this graph as text
The chain rule composes two local scale changes. A small input change h is first scaled by g'(a) , then the resulting change is scaled by f'(g(a)) . The total first-order scale factor is their product, in the same order as the function composition. Follow the perturbation from left to right. The inner function converts h into approximately g'(a)h . The outer function acts on that new change, producing approximately f'(g(a))g'(a)h . Multiplication appears because two one-dimensional linear scalings are composed.
The visual uses labeled positions, solid and dashed line styles, and written descriptions so the chain rule composes two local scale changes does not depend on color.
Why it matters: This diagram reframes the chain rule as a statement about local input-output transformations rather than a mnemonic about "outside" and "inside." It prepares advanced readers for Jacobian matrices while remaining intelligible in one dimension.
A small input change h is first scaled by g'(a), then the resulting change is scaled by f'(g(a)). The total first-order scale factor is their product, in the same order as the function composition.
In higher dimensions, and become matrices representing local linear maps. The chain rule becomes
with matrix order reflecting the order of composition.
Automatic differentiation and computational graphs
Modern scientific software and machine-learning systems organize a calculation as a graph of elementary operations. Derivatives are propagated through that graph by repeated chain-rule applications. Forward mode tracks how one input perturbation moves through the computation; reverse mode propagates output sensitivity backward and is the foundation of backpropagation.
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