∇Continuous Function
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Derivatives

The derivative is an instantaneous rate of change: the slope you get when a secant line becomes a tangent line.

published · difficulty 2/5 · 14 min read

01

01

Intuition

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

PredictName the object in plain language, then predict what should change.Leave with one reusable mental picture before notation appears.

The derivative is the “instantaneous slope.”

If you stand on a curve and look a tiny step to the right, the secant line gives you an average slope. As that step shrinks to zero, the average slope becomes the local slope: the tangent line.

In optimization, this local slope is the gradient: it tells you how to change parameters to decrease a loss.

Section prompt

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02

02

Math

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

InspectTrack the same object through the notation and check each symbol.Leave with the invariant the equations preserve.
Equation 1Inspect equation objectf'(x) = \lim_{h \to 0} (f(x+h) - f(x))/(h).Equation 2Inspect equation object(f(x+h) - f(x))/(h)Equation 3Inspect equation objecty = f(x_0) + f'(x_0)(x - x_0).

Definition (derivative). The derivative of fff at xxx is

f′(x)=lim⁡h→0f(x+h)−f(x)h.f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.f′(x)=h→0lim​hf(x+h)−f(x)​.

The fraction

f(x+h)−f(x)h\frac{f(x+h) - f(x)}{h}hf(x+h)−f(x)​

is a secant slope (average rate of change over an interval). The limit turns it into a tangent slope.

Once you know f′(x0)f'(x_0)f′(x0​), the tangent line at x0x_0x0​ is

y=f(x0)+f′(x0)(x−x0).y = f(x_0) + f'(x_0)(x - x_0).y=f(x0​)+f′(x0​)(x−x0​).
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03

03

Code

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

TraceMatch variables to symbols before reading the implementation.Leave with a runnable witness for the math.
Code witness 1import numpy as np f = lambda x: np.sin(x) df_true = lambda x: np.cos(x) x0 = 1.2 for h in [1...python
import numpy as np

f = lambda x: np.sin(x)
df_true = lambda x: np.cos(x)

x0 = 1.2
for h in [1.0, 0.3, 0.1, 0.03, 0.01]:
    df_est = (f(x0 + h) - f(x0)) / h
    print("h=", h, "secant=", df_est, "true=", df_true(x0))
Section prompt

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04

04

Interactive Demo

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

ManipulateChange one control and predict the visible response before reveal.Leave with the observed invariant or a repaired model.

Live Concept Demo

Explore Derivatives

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

difficulty 2/5undergraduatecode-aligned
Demo inquiry checkpoint

Manipulate one control and predict the visible change.

01Choose lensTrace a quantity
02ObserveDemo state pending
03GroundName the equation, invariant, or control that explains it.
04CarryNext: Gradient Descent

Choose what to inspect in Derivatives. This shared fallback is an observation guide, not evidence of learning.

Loading interactive demo...

Use the demo to inspect the visible secant slope, predict the hidden tangent relation, then reveal how the derivative appears as hhh shrinks.

Section prompt

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4/4 sections ready

Concept: Derivatives

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

BeforeFunctionsNow4/4 sections readyTryManipulate one control and predict the visible change.NextGradient Descent
Object contextCalculus
StateCalculus
1 source attachedChecking local snapshot
concept:calculus/derivatives
Math3 equation objectsf'(x) = \lim_{h \to 0} (f(x+h) - f(x))/(h).CodeDerivatives code witness 1f = lambda x: np.sin(x)DemoDerivatives predictionManipulate one control and predict the visible change.
ConceptLearner lens

Derivatives

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

Mode questionCan I say the mechanism back in one sentence before I reveal anything?

Start with the prediction checkpoint, then compare the reveal to the mental model.

Take this move

Study modes

Keep the object fixed; change the lens.
01PredictCommit before reveal.02ExplainName the invariant.03Ground1 source link nearby.04ExperimentChange one knob.05CarrySave the next question.

Route back through the notebook

Carry the same object through intuition, math, code, and demo.

4/4 sections ready
Carry inFunctions

Bring the mental model from Functions; this page will reuse it instead of restarting from zero.

Work hereDerivatives

The derivative is an instantaneous rate of change: the slope you get when a secant line becomes a tangent line.

Carry outGradient Descent

The next edge should feel earned: use the demo prediction here before following Gradient Descent.

Test the linkManipulate one control and predict the visible change.Then continue to Gradient Descent

After The First Pass

Turn the concept into an inspected object.

The lower panels are one second act: keep the object fixed, inspect it visually, check source boundaries, practice transfer, then attach the research question.
ConceptDerivativesCalculus
01MechanismInspect motionSee how the idea changes form before adding more explanation.02VisualChoose a cueBind one visible mark to one mathematical role.03SourcesCheck boundary1 source link attached to the object.04ClaimAudit evidenceTreat the page claim as something to verify.05PracticeAttempt transferWrite before feedback so the model has something to repair.06RoomAsk groundedAttach the question to the same exact object.

Mechanism Storyboard

See the idea move before the page explains it

The derivative is an instantaneous rate of change: the slope you get when a secant line becomes a tangent line.

Demo notes open01 / Intuition
Editorial calculus illustration of a curve, secant line, tangent slope, and local rate-of-change marker.
Prediction lens

Start with the picture, metaphor, or geometric mechanism.

Commit first

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

Visual Inquiry

Make the image answer a mathematical question

The derivative is an instantaneous rate of change: the slope you get when a secant line becomes a tangent line.

4/4 stages readyDemo notes connected
Prediction

Which visible object should carry the first intuition?

Commit first

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

Source Grounding

Canonical references for the mechanism on this page.

Object - ConceptDerivativesQuestion

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

concept:calculus/derivatives
Boundary

sources: deisenroth-2020-mml

Check

Open the closest source note before trusting the local explanation.

Evidence

1 selected-object source shown first; 1 reference total.

Next move

Audit the claim boundary, then ask from the same selected object.

selected object source · book · 2020Mathematics for Machine LearningDeisenroth, Faisal, and Ong
Located CF editorial boundary

Grounds derivatives and vector calculus as part of the mathematical toolkit needed for machine learning.

Used here as

MML's vector calculus chapter defines the derivative as the limit of a difference quotient, explains the secant-to-tangent picture for differentiable functions, and notes derivatives/grad...

Caveat

Checks differentiable one-variable real functions and positive/right-hand h values in the toy demo. Optimization is only the one-dimensional instance of gradient direction; not corners/cu...

Open source

Claim Review

The derivative is an instantaneous rate of change: the slope you get when a secant line becomes a tangent line.

Object - ConceptDerivativesQuestion

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

concept:calculus/derivatives
Boundary

sources: deisenroth-2020-mml

Check

Treat every claim as provisional until source support and a local witness agree.

Evidence

1 structured claim check on this concept.

Next move

Run the prediction or practice transfer before asking for a grounded review.

1 CF editorial source-scope review recorded

Publisher-side editorial review is not independent replication. Claims without it still need exact source-support review. 1 reference and 3 local witnesses are available for inspection.

For a differentiable one-variable real function, the derivative at x is the limit of the difference quotient as h approaches 0: secant slopes over [x,x+h] converge to the tangent slope, a local one-variable rate-of-change signal used by gradient-based optimization.
Registered CF editorial boundary
Mathematics for Machine LearningGrounds derivatives and vector calculus as part of the mathematical toolkit needed for machine learning.
Used here as

MML's vector calculus chapter defines the derivative as the limit of a difference quotient, explains the secant-to-tangent picture for differentiable functions, and notes derivatives/gradients give steepest-...

Local witness
source-span-deisenroth-2020-mmlmath-object-1math-object-2code-witness-1interactive-demo
Equation 1
f′(x)=lim⁡h→0f(x+h)−f(x)h.f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.
Equation 2
f(x+h)−f(x)h\frac{f(x+h) - f(x)}{h}
Caveat

Checks differentiable one-variable real functions and positive/right-hand h values in the toy demo. Optimization is only the one-dimensional instance of gradient direction; not corners/cusps, subgradients, p...

Review stateCF editorial source-scope reviewClaim metadata: source checkedPublisher-side editorial review only; not independent replication. Check caveats and exact source scope.

Checked MML 5.1 and 7.1: MML defines the difference quotient for a univariate real function, identifies it as the secant/average slope between x and x+delta x, and says its limit becomes the tangent/derivative when f is differentiable. MML then uses gradients of differentiable scalar objectives as steepest-ascent signals for gradient descent. Local math/code/demo match the one-variable specialization: f'(x)=lim (f(x+h)-f(x))/h, positive-h secants for sin vs cos, and secant-vs-hidden-tangent reveal.

Reviewer: codex+oracle; reviewed 2026-05-07

Practice notebook

Use the idea, then test it somewhere new

The derivative is an instantaneous rate of change: the slope you get when a secant line becomes a tangent line.

AttemptNo learning claim inferred
Object - ConceptDerivativesQuestion

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

concept:calculus/derivatives
Boundary

sources: deisenroth-2020-mml

Check

Use one state from Derivatives to explain what changes, why it changes, and which assumption the explanation needs.

Evidence

No learner move yet; no learning state is inferred.

Next move

Write first, use only the help you need, then try a new case without it.

Explain

Use one state from Derivatives to explain what changes, why it changes, and which assumption the explanation needs.

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.

Grounded object roomClose
Selected object routeAsk from this object; carry one invariant back.sources: deisenroth-2020-mml
  1. ObjectConceptDerivatives
  2. PredictBefore revealDerivatives prediction
  3. WitnessCompare codeDerivatives code witness 1
  4. RoomAsk groundedChecking local snapshot
ConceptDerivativesCalculus
Code witness comparisonDerivatives code witness 1f = lambda x: np.sin(x)Prediction before revealDerivatives predictionManipulate one control and predict the visible change.
Grounded room questionWhat is the smallest example that makes Derivatives click without losing the math?Checking local snapshot

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

Derivatives

Anchored question

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

Source boundaryInspect source ids: deisenroth-2020-mmlStable content-object key attached
Role lenses for this object

These are fixed, deterministic perspectives derived from the selected object. They do not represent people, community contributions, or independent review.

Learner evidence requestAsk what would make "Derivatives" feel predictable rather than familiar.
Assumption

Source ids deisenroth-2020-mml must support the exact object, not just the surrounding topic.

Source-checking summary

Treat this as a mechanism object: connect the definition to one equation, code witness, or demo before broadening the discussion.

Proposed experiment

Ask the learner to perturb one representation, then check whether the same invariant survives in math, code, and demo.

Next action

The learner can state the mechanism in their own words

Evidence4 checks
PredictionChecking carried observation
ActionReady for one action
AILearner handoff ready
Open source object
01PredictionChecking browser-local route memory
02EvidenceChecking for a carried observation
03BoundaryInspect source ids: deisenroth-2020-mml
04Next moveSave one next action
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/derivatives.

No local draft saved.
Evidence to inspect
  • Source ids to inspect: deisenroth-2020-mml
  • 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
Object-attached AI handoff

I am working in Continuous Function's research reading room. Object: concept - Derivatives Object key: concept:calculus/derivatives Context: Calculus Anchor id: concept/concept-notebook/calculus/derivatives Open question: What is the smallest example that makes Derivatives click without losing the math? Evidence to inspect: - Source ids to inspect: deisenroth-2020-mml - 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 Deterministic role lenses for this object: - Boundary: fixed perspectives, not people, community contributions, or independent review - Source-checking summary: Treat this as a mechanism object: connect the definition to one equation, code witness, or demo before broadening the discussion. - Proposed experiment: Ask the learner to perturb one representation, then check whether the same invariant survives in math, code, and demo. - Teach/transfer move: Turn the mechanism into one sentence that predicts a neighboring concept. - Assumptions: - Source ids deisenroth-2020-mml must support the exact object, not just the surrounding topic. - The stable content-object key lets local drafts, prompts, and route memory attach without changing the source page. - The concept explanation is local atlas prose until checked against its math, code, and source support. - Prerequisite gaps should become a repair route, not a reason to leave the object vague. - Role-lens requests: - Learner: ask for "Ask what would make "Derivatives" feel predictable rather than familiar." | assumption: Source ids deisenroth-2020-mml must support the exact object, not just the surrounding topic. | next action: The learner can state the mechanism in their own words - Researcher: ask for "Source ids to inspect: deisenroth-2020-mml" | assumption: The stable content-object key lets local drafts, prompts, and route memory attach without changing the source page. | next action: The learner can name the prerequisite that would repair confusion - Experimenter: ask for "Choose one variable or condition to perturb before asking for an explanation." | assumption: The concept explanation is local atlas prose until checked against its math, code, and source support. | next action: The learner can predict how the mechanism changes under one perturbation - Professor: ask for "Find the smallest transferable rule a learner could reuse without the AI." | assumption: Prerequisite gaps should become a repair route, not a reason to leave the object vague. | next action: Teach or transfer: Turn the mechanism into one sentence that predicts a neighboring concept. 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. Current deterministic role lens for this object: - Role lens: Learner - Evidence request: Ask what would make "Derivatives" feel predictable rather than familiar. - Assumption to keep visible: Source ids deisenroth-2020-mml must support the exact object, not just the surrounding topic. - Proposed experiment: Ask the learner to perturb one representation, then check whether the same invariant survives in math, code, and demo. - Next action: The learner can state the mechanism in their own words

concept/concept-notebook/calculus/derivatives concept:calculus/derivatives