Calculus

Chain Rule

Derivatives of composed functions multiply: small changes propagate through a chain of dependencies.

status: reviewimportance: criticaldifficulty 2/5math: undergraduateread: 6mlive demo

Back to Calculus Next: Computation Graphs

Concept Structure

Chain Rule

01Intuition

Start with the picture, metaphor, or geometric mechanism.

02Math

Make the objects explicit and connect them with notation.

03Code

Mirror the equations with runnable implementation details.

04Interactive Demo

Manipulate the mechanism and watch the idea respond.

1prerequisites

1next concepts

2related links

Learner Contract

What this page should let you do.

You are here becauseDerivatives of composed functions multiply: small changes propagate through a chain of dependencies.

This Calculus concept is the current object: keep the same invariant visible across Intuition, Math, Code, Interactive Demo.

Before thisDerivatives

1 prerequisite listed; refresh them before leaning on the math or code.

Derivatives

By the end4/4 sections ready | code witness expected | live demo

Explain the mechanism, trace the main notation, and test one prediction in the live demo.

Do this firstIntuition

Read the intuition before the notation; the math should name a mechanism you already felt.

Then go nextComputation Graphs

Follow this edge after making one prediction here; the next page should reuse the result, not restart the route.

Test the linkManipulate one control and predict the visible change.Then continue to Computation Graphs

Claim/source review status

Claim review not recorded

No structured claim checks yet; source metadata is not enough to establish support.Metadata-derived; review may be AI-assisted. Not a human certification.

Claims0/0 reviewed

Sources0 cited

Codeattached

Demolive

Reviewednot recorded

Updatedpage 2026-06-29

Object flow

4/4 sections readyAsk about this Research room

ConceptChain RuleCalculus EquationChain Rule equation 1Exact equation object CodeChain Rule code witness 1Exact code witness DemoChain Rule interactive demoVisualization object ClaimChain Rule central claimMechanism claim EquationChain Rule math objectsEquation and notation

ConceptChain RuleCalculus

Local snapshot ready

concept:calculus/chain-rule

Codewitness nearby Predictbefore reveal Roomobject handoff

01IntuitionStart with the picture, metaphor, or geometric mechanism.02MathMake the objects explicit and connect them with notation.03CodeMirror the equations with runnable implementation details.04Interactive DemoManipulate the mechanism and watch the idea respond.

Intuition

Build the mental picture first so the rest of the page has something to attach to.

Section prompt

When a quantity depends on an intermediate variable, change propagates through the chain.

If a tiny change in $x$ causes a change in $u$ , and that change in $u$ causes a change in $y$ , then the total change from $x$ to $y$ multiplies those effects.

That is the entire story of backpropagation: derivatives flow through a computational graph by repeatedly applying the chain rule.

Math

Translate the story into symbols, assumptions, and a derivation you can inspect.

Section prompt

Equation 1 $(f \circ g)'(x) = f'(g(x))\, g'(x).$

Let $u = g(x)$ and $y = f(u) = f(g(x))$ . The chain rule says:

(f \circ g)'(x) = f'(g(x))\, g'(x).

A useful way to remember it is “differentiate the outside, then multiply by the derivative of the inside.”

For multi-variable functions (what we use in ML), the same idea becomes matrix multiplication between local Jacobians. The later autodiff pages turn this into forward and reverse passes; here, the important invariant is just that local rates multiply along a dependency chain.

Code

Keep the implementation aligned with the notation so the algorithm is legible.

Section prompt

Code witness 1import torch x = torch.tensor(1.5, requires_grad=True) # y = f(g(x)) with g(x)=x^2 and f(u)=s...python

import torch

x = torch.tensor(1.5, requires_grad=True)

# y = f(g(x)) with g(x)=x^2 and f(u)=sin(u)
y = torch.sin(x**2)
y.backward()

print("dy/dx (autograd):", x.grad.item())

# Analytic chain rule: f'(u)=cos(u), g'(x)=2x
analytic = torch.cos(x**2) * (2 * x)
print("dy/dx (analytic):", analytic.item())

Interactive Demo

Use direct manipulation to connect the explanation to a moving system.

Section prompt

Use the live witness below: the forward values are visible, but the local rates are hidden. Predict whether a tiny positive move in $x$ raises, lowers, or barely changes $y$ for

u=x^2,\qquad y=\sin(u).

Then reveal $du/dx$ , $dy/du$ , their product $dy/dx$ , a forward finite-step check, and a central-slope check. Try the presets after the default case: one link can flip the sign, and one nearly stopped link can gate the first-order chain even when a finite step still shows a tiny second-order change.

Live Concept Demo

Explore Chain Rule

The stage is code-native and interactive. Use it to test the explanation against the mechanism.

difficulty 2/5undergraduatecode-aligned

Demo Prediction Checkpoint

Manipulate one control and predict the visible change.

Commit to what Chain Rule should make visible before reading the result.

After The First Pass

Turn the concept into an inspected object.

Once the invariant is visible in the intuition, math, code, and demo, use these panels to inspect the mechanism visually, check source support, practice the idea, and attach a grounded research question.

Mechanism Storyboard

See the idea move before the page explains it

Derivatives of composed functions multiply: small changes propagate through a chain of dependencies.

Prediction open01 / Intuition

Prediction lens

Start with the picture, metaphor, or geometric mechanism.

Commit first

Before reading further, choose the kind of change Chain Rule should make visible.

Visual Inquiry

Make the image answer a mathematical question

Derivatives of composed functions multiply: small changes propagate through a chain of dependencies.

4/4 stages readyLive demo connected

Visual cueWhich visible object should carry the first intuition?

Inspection depth2/4

Prediction

Which visible object should carry the first intuition?

Commit first

Pick the cue that should make Chain Rule easier to reason about before the page gives the answer.

Source Grounding

Canonical references for the mechanism on this page.

Source gapNo canonical references are listed yet.

Add source metadata before treating this page as source-grounded.

Claim Review

Derivatives of composed functions multiply: small changes propagate through a chain of dependencies.

StatusSubstantive claim review pending

Source IDs and witness objects are attached for review; they are not proof by themselves.

SourcesNo references

Add source metadata before claiming support.

Witnesses3 local objects

Use equation, code, and demo objects to check whether the source support is operational.

Source support candidates

No structured source note is attached yet.

Mechanism witnesses

Equation 1

(f \circ g)'(x) = f'(g(x))\, g'(x).

Code witness 1import torch x = torch.tensor(1.5, requires_grad=True) # y = f(g(x)) with g(x)=x^2 and f(u)=s...Demo stateLive mechanism probe

Practice Loop

Try the idea before it explains itself

Derivatives of composed functions multiply: small changes propagate through a chain of dependencies.

Readiness0/3 checks ready

Predict

Before touching the demo, predict one visible change that should happen in Chain Rule.

Hint 1

Reveal when your model needs a nudge.

Hint 2

Reveal when your model needs a nudge.

Hint 3

Reveal when your model needs a nudge.

Your answer canvas

Local checks

Claim

A concrete answer is on the canvas.

Mechanism

The answer names why the claim should hold.

Bridge

It touches the page context or a neighboring idea.

Misconception check

Object research drawerClose

ConceptChain RuleCalculus

Code witness comparisonChain Rule code witness 1x = torch.tensor(1.5, requires_grad=True)Prediction before revealChain Rule interactive demoManipulate one control and predict the visible change.

Grounded room questionWhat is the smallest example that makes Chain Rule click without losing the math?Local snapshot ready

Research Room

Attach the question to an exact object

Pick the concept, equation, source, code witness, claim, misconception, or demo state before asking for help. The handoff stays grounded to that object.

Next local actionNo local draft saved yet

Open the draft below to save one note and next action in this browser.

conceptCalculus

Chain Rule

Anchored question

What is the smallest example that makes Chain Rule click without losing the math?

Local action draftNo local draft saved yetExpand only when ready to capture one local next action

Local action draft

This draft stays locally in this browser for concept:calculus/chain-rule.

Draft noteNext action

No local draft saved.

Evidence to inspect

Definition, prerequisite, and contrast concept links
The equation or code witness that makes the concept operational
One demo state that shows the invariant instead of a slogan

What would resolve this

The learner can state the mechanism in their own words
The learner can name the prerequisite that would repair confusion
The learner can predict how the mechanism changes under one perturbation

Grounded AI handoff

I am working in Continuous Function's research reading room. Object: concept - Chain Rule Object key: concept:calculus/chain-rule Context: Calculus Anchor id: concept/concept-notebook/calculus/chain-rule Open question: What is the smallest example that makes Chain Rule click without losing the math? Evidence to inspect: - Definition, prerequisite, and contrast concept links - The equation or code witness that makes the concept operational - One demo state that shows the invariant instead of a slogan What would resolve this: - The learner can state the mechanism in their own words - The learner can name the prerequisite that would repair confusion - The learner can predict how the mechanism changes under one perturbation Answer as a careful research tutor: stay source-grounded, separate verified evidence from assumptions, name the relevant math objects, and end with one next action.

Open source object

concept/concept-notebook/calculus/chain-rule
concept:calculus/chain-rule

Learning Map

Before / Now / Try / Next

BeforeDerivatives

NowIntuition → Math → Code → Demo

TryManipulate one control and predict the visible change.

NextComputation Graphs

Intuitionready
Mathready
Codeready
Interactive Demoready

Object Companion

Ask beside the selected object

Derivatives of composed functions multiply: small changes propagate through a chain of dependencies.

Your question

GoalComfortStyleStuck on

Context prompt

You are my AI learning companion for Continuous Function. Current context: Calculus concept. Learning surface: Chain Rule. What this page says: Derivatives of composed functions multiply: small changes propagate through a chain of dependencies. Current section: Intuition, math, code, and interactive demo. Suggested next step: Manipulate one control and predict the visible change.. Learner goal: Understand the idea. Learner comfort level: New to this. Preferred explanation style: Visual first. Task: Explain the central idea in plain language, then restate it with the exact math objects from the page. Answer in a way that helps me learn: ask one clarifying question only if needed, use intuition before notation, and end with one thing I should try on the page.

Domain

Calculus

calculusderivativescomposition

Prerequisites

Derivatives

Leads To

Computation Graphs

Reverse-Mode Automatic Differentiation Backpropagation

Within this domain