Math · Calculus

Derivative as Slope / Local Rate of Change

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What you learned

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Read next: Limits and Continuity / Approaching a ValueSupport the limit picture underneath derivative

Key takeaway

The derivative at one point is the tangent slope there: the local rate of change on the original curve.
Shrinking delta x makes the secant slope approach that tangent slope.
The derivative graph stores the local tangent slope across the whole curve.

Common misconception

The derivative is just another average slope between two nearby points.

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

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Equation snapshotSecant slope · DerivativeShow 3 equations

Secant slope
$secant slope = \frac{f ( x + Δ x ) - f ( x )}{Δ x}$
Measures the average rate of change across a finite horizontal interval.
Derivative
$f^{'} (x) = lim_{Δ x \to 0} \frac{f ( x + Δ x ) - f ( x )}{Δ x}$
Defines the tangent slope as the value the secant slope approaches when Δx shrinks to 0.
Tangent line
$y - f (a) = f^{'} (a) (x - a)$
Uses the derivative at one point to write the line with the same local slope through that point.

Why it behaves this way

Explanation

The derivative is the slope the curve has at one chosen point. This module keeps the point, the tangent line, the secant line, and the derivative graph linked together so you can see that local slope on the original curve before reading it on a second graph.

A secant uses two points and gives an average rate of change across an interval of width Δx. As Δx shrinks, the second point moves toward the first, the secant slope approaches the tangent slope, and that limiting local slope is the derivative.

Key ideas

01The derivative at one point is the tangent slope there: the local rate of change on the original curve.

02A secant slope is an average rate of change across a finite interval, so shrinking Δx moves that average toward the tangent slope.

03The derivative graph records those tangent slopes as the point moves along the original curve.

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Frozen walkthrough

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Frozen walkthrough

Use the current point and Δx on the live curve. First read the secant as an average rate over an interval, then compare it with the tangent and the derivative graph.

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Example 1 of 2

Frozen valuesUsing frozen parameters

For the current point and Δx, what average rate of change does the secant line represent over that interval?

x

Point position

Δ x

Delta x

0.8

1. Read the two points on the curve

The secant runs from

(0, 0)

(0.8, - 0.71)

2. Form the difference quotient

Use

\frac{Δ y}{Δ x} = \frac{- 0.71 - 0}{0.8}

, where

Δ y = - 0.71

3. Compute the average rate

So the secant slope is

\frac{- 0.71}{0.8} = - 0.88

Current average rate

m_{secant} = - 0.88

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.

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Step 4 of 6

Functions and Change

Derivative as Slope / Local Rate of Change appears later in this track, so it is cleaner to start from the beginning first.

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