≥Better Grades
Menu

Calculus I · Unit 2A · exploration

The Chain Rule as Composition of Local Linear Maps

Why the Derivative Factors Multiply

Write local models

g(a+h)=g(a)+g′(a)h+o(h)g(a+h)=g(a)+g'(a)h+o(h)

and

f(g(a)+k)=f(g(a))+f′(g(a))k+o(k).f(g(a)+k)=f(g(a))+f'(g(a))k+o(k).

Substitute the change in gg for kk. To first order,

f(g(a+h))−f(g(a))=f′(g(a))g′(a)h+o(h).f(g(a+h))-f(g(a)) =f'(g(a))g'(a)h+o(h).

The coefficient of hh is the chain-rule product.

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.
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.

Visual study

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, g′(a)g'(a) and f′(g(a))f'(g(a)) become matrices representing local linear maps. The chain rule becomes

D(f∘g)(a)=Df(g(a)) Dg(a),D(f\circ g)(a)=Df(g(a))\,Dg(a),

with matrix order reflecting the order of composition.

Application

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.

After the explanation

Use the section idea

Reading lens

Use these optional explorations to see the deeper analysis behind familiar derivative rules.

Mental model

Differentiability is a local linear approximation property with consequences beyond computation.

Decision

Return here after the core path is secure, and connect each abstraction to a concrete derivative example.

Common trap

Collecting formal language without linking it to the local linear model it describes.

Check yourself

Can you restate the advanced claim in ordinary language and test it on an example?

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.