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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.

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Starter track

Step 4 of 60 / 6 complete

Functions and Change

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

1. Graph Transformations2. Rational Functions / Asymptotes and Behavior3. Exponential Change / Growth, Decay, and Logarithms4. Derivative as Slope / Local Rate of Change+2 more steps

Previous step: Exponential Change / Growth, Decay, and Logarithms.

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Why it behaves this way

Explanation

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.

Frozen walkthrough

Step through the frozen example

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

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Frozen valuesUsing frozen parameters

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

Point location

-1.2

Delta x

0.8

1. Read the two points on the curve

The secant runs from to .

2. Form the difference quotient

Use , where .

3. Compute the average rate

So the secant slope is .

Current secant slope

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.

Make a prediction before you reveal the next step.

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

Check your reasoning against the live bench.

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?

Use the live bench to test the result before moving on.

Accessibility

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.