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MathCalculusIntroStarter track

Concept module

Derivative as Slope / Local Rate of Change

Slide a point along one curve, tighten a secant into a tangent, and connect local steepness to the derivative graph without leaving the same live bench.

Interactive lab

Keep the stage, graph, and immediate control feedback in one working view.

Derivative as slope

Drag the curve point or secant handle to inspect how local slope changes.

Graphs

Switch graph views without breaking the live stage and time link.

Derivative graph

Shows how the tangent slope changes across the full curve.

x: -4.41 to 4.41f'(x): -1.62 to 7.42

f'(x)

Hover or scrub to link the graph back to the stage.x / f'(x)

Controls

Adjust the physical parameters and watch the motion respond.

Point position

x

-1.2

Move the active point along the curve.

Delta x

Δ x

0.8

Set the secant-point spacing used for the average rate.

Show secant

s

Hide or show the secant comparison line.

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

ObservationPrompt 1 of 3

Move the point left and right and watch the tangent line switch from falling to rising as the derivative crosses zero.

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

x

Point position

-1.2

Slides the active point along the curve, which changes the point value, the tangent slope, and the derivative-graph reading together.

Graph: Derivative graphGraph: Secant slope vs delta xOverlay: Tangent lineOverlay: Delta guide

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Use the live prompts to keep the geometry and the graph reading tied together.

ObservationPrompt 1 of 3

Graph: Derivative graph

Move the point left and right and watch the tangent line switch from falling to rising as the derivative crosses zero.

Control: Point positionGraph: Derivative graphOverlay: Tangent lineEquation

x

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

3 visible

Overlay focus

Tangent line

Show the instantaneous slope at the current point.

What to notice

The tangent touches the curve at one point and matches its local direction there.

Why it matters

It makes the derivative visible as a slope, not just as a symbol.

Control: Point positionGraph: Derivative graphEquation

x

Challenge mode

Use the live point and secant interval as a compact slope bench. The goal is to make the tangent and secant tell the same near-flat story for a real reason, not just to hunt a number.

0/1 solved

ConditionCore

3 of 8 checks

Catch the flat tangent

Move to the right-hand turning point so the tangent is almost flat, then shrink delta x until the secant slope is almost flat there too.

Graph-linkedGuided start2 hints

Suggested start

Use the secant graph and the delta guide together while you move toward the right-hand turning point.

Pending

Open the Secant slope vs delta x graph.

Derivative graph

Matched

Keep the Tangent line visible.

Matched

Keep the Secant line visible.

Matched

Keep the Delta guide visible.

Pending

Move the active point into the right-hand flat-turn band between

1.25

and

1.45

-1.2

Pending

Shrink

Δ x

into a tight local interval between

0.15

and

0.3

0.8

Pending

Bring the tangent slope into the near-zero band from

- 0.08

0.08

-0.22

Pending

Keep the secant slope close too, between

- 0.12

and

0.12

-0.63

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

At x = -1.2, the point on the curve is y = 0.89 and the tangent slope is -0.22. The tangent falls from left to right here. With delta x = 0.8, the secant slope is -0.63.

Equation detailsDeeper interpretation, notes, and worked variable context.

Secant slope

Measures the average rate of change across a finite horizontal interval.

secant slope = \frac{f ( x + Δ x ) - f ( x )}{Δ x}

x

Point position -1.2

Δ x

Delta x 0.8

s

Show secant On

Derivative

Defines the tangent slope as the limiting value of the secant slope.

f^{'} (x) = Δ x \to 0 lim \frac{f ( x + Δ x ) - f ( x )}{Δ x}

x

Point position -1.2

Δ x

Delta x 0.8

s

Show secant On

Tangent line

Uses the derivative at one point to write the tangent line through that point.

y - f (a) = f^{'} (a) (x - a)

x

Point position -1.2

Progress

Not startedMastery: NewLocal-first

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Let the live model runChange one real controlOpen What to notice

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Stable links

Starter track

Step 2 of 30 / 3 complete

Functions and Change

Earlier steps still set up Derivative as Slope / Local Rate of Change.

1. Graph Transformations2. Derivative as Slope / Local Rate of Change3. Integral as Accumulation / Area

Previous step: Graph Transformations.

Start from track beginning

Short explanation

What the system is doing

The derivative stops feeling abstract when the slope question stays attached to one visible point on one visible curve. This module keeps the point, the tangent line, the secant line, and the derivative graph in view together so local rate of change never drifts into detached notation.

The curve here is fixed on purpose. You move the point along it, shrink or widen delta x, and watch the average rate over a secant settle toward the instantaneous slope of the tangent.

Key ideas

01A secant slope is an average rate of change between two nearby points on the curve.

02A tangent slope is the local rate of change at one point, and it is what the derivative records.

03The derivative graph shows how that local slope changes as you slide the point across the curve.

Worked example

Read the full frozen walkthrough.

Frozen walkthrough

Use the current point and secant on the live curve. The same controls drive the stage, the graph tabs, and these substitutions.

Live worked examples are available on Premium. You can still read the full frozen walkthrough on the free tier.

View plans

Frozen valuesUsing frozen parameters

For the current point and delta x, what average rate of change does the secant line show?

x

Point location

-1.2

Δ x

Delta x

0.8

1. Read the two points on the curve

The secant runs from

(- 1.2, 0.89)

(- 0.4, 0.39)

2. Form the difference quotient

Use

\frac{Δ y}{Δ x} = \frac{0.39 - 0.89}{0.8}

, where

Δ y = - 0.5

3. Compute the average rate

So the secant slope is

\frac{- 0.5}{0.8} = - 0.63

Current secant slope

m_{secant} = - 0.63

The secant slope still differs noticeably, which is the cue that Δx has not shrunk enough yet for the average rate to match the local one closely.

Common misconception

The derivative is a second graph that has nothing to do with the original curve once the formula is written down.

The derivative graph is built from the slopes of tangent lines on the original curve.

When the tangent is steep and rising, the derivative value is positive and large. When the tangent is flat, the derivative value is near zero. When the tangent falls, the derivative value is negative.

Mini challenge

Move the point until the tangent is rising but the secant slope is still noticeably different from the tangent slope.

Prediction prompt

Decide whether you need a larger or smaller delta x before you test it.

Check your reasoning

You need a point where the tangent slope is positive and a delta x that is still large enough for the secant to show a visibly different average rate.

The tangent slope depends on the local steepness at one point, while the secant slope averages over an interval. Keeping delta x larger preserves that gap.

Quick test

Graph reading

Question 1 of 3

Use the tangent, secant, and derivative graph together. These checks are about how slope behaves across the curve.

If the tangent line at a point is horizontal, what must the derivative be there?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

A single coordinate grid shows the original curve, one movable point on the curve, an optional secant point, and the tangent line at the main point. Optional guides show the horizontal and vertical changes used in the difference quotient.

Dragging the main point changes the local slope and the derivative reading together. Dragging the secant point changes the interval used for the average rate.

Graph summary

The derivative graph plots tangent slope against x, so it acts as a slope map for the original curve.

The difference-quotient graph plots secant slope against delta x, so it shows how the average rate approaches the tangent slope as the interval shrinks.

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Carry local slope into running totals and signed areaCalculus

Derivative as Slope / Local Rate of Change

See each variable before you move it.

Tangent line

Catch the flat tangent

Functions and Change

What the system is doing

Read the full frozen walkthrough.

For the current point and delta x, what average rate of change does the secant line show?

If the tangent line at a point is horizontal, what must the derivative be there?

Keep moving through the concept map.

Integral as Accumulation / Area

Graph Transformations

Vectors in 2D