AP Calculus AB · Lesson review · September 4, 2026

Graphing the Derivative

Every tangent slope becomes a height on a new graph. Start with horizontal tangents, follow the signs, and look carefully at corners and vertical tangents.

33-minute edited lesson12 chaptersCaptions + corrected transcript

01 · Watch

Lesson video

VTT captions SRT captions

0:0001Graphing the Derivative 1:5702Start with Zeros, Signs, and Steepness 3:1103A Smooth Graph and Polynomial Degree 8:2604Parabolas and a Reciprocal-Type Graph 12:3205Endpoints and Restricted Domains 16:4206Corners and One-Sided Slopes 19:1907Cosine Becomes Negative Sine 21:3708The Exponential That Matches Its Derivative 24:1609Absolute Value Creates Corners 28:0210Piecewise-Linear Graphs 29:3211Cube Roots and Vertical Tangents 31:5212Sketching a Semicircle Derivative

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02 · Understand

Learning targets

Interpret $f'(a)=m_{\mathrm{tangent}}$ : keep the input, but replace the function height with its tangent slope.
Locate horizontal tangents and plot zeros of the derivative.
Use slope signs and relative steepness to sketch between those zeros.
Distinguish a horizontal tangent from a corner, endpoint, or vertical tangent.
Recognize derivative graphs for familiar families and explain where a derivative does not exist.

03 · Vocabulary

A slope-reading toolkit

A derivative graph records slopes, not the original function values.

01Derivative at a point: The limiting secant slope, when the limit is finite: $f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$ .
02Horizontal tangent: The tangent has zero slope, so $f'(a)=0$ . This can happen at a smooth maximum or minimum, but not every zero must be an extremum.
03Sign and steepness: A positive tangent slope gives $f'(x)>0$ ; a negative tangent slope gives $f'(x)<0$ . A steeper tangent corresponds to a larger derivative magnitude.
04Corner: The incoming and outgoing tangent slopes disagree. A function can be continuous while its derivative is undefined.
05Vertical tangent: A vertical tangent has no finite slope, so there is no finite derivative at that input.
06Endpoint convention: This lesson uses the ordinary two-sided derivative: at a boundary of the domain, one side is unavailable. A one-sided derivative may still exist and must be identified as one-sided.

04 · Apply

From curve to derivative

Most classroom graphs are qualitative sketches. Preserve the sign, zeros, and changes in steepness; do not invent precise coordinates.

Opening example

A parabola becomes a line

f(x)=x^2-3x\quad\Longrightarrow\quad f'(x)=2x-3

The parabola has a horizontal tangent at $x=\frac32$ .
To its left, tangent slopes are negative; to its right, they are positive.
The slopes increase steadily, producing an increasing derivative line.

The general strategy also works without a formula: locate zeros, decide signs, and compare steepness.

A smooth graph

Zeros first, then signs

Mark every horizontal tangent on the original curve.
At those same inputs, put the derivative on the horizontal axis.
Between them, decide whether tangent slopes are positive or negative.
Place the derivative farther from zero where the curve is steeper, then connect with a plausible smooth shape.

Polynomial extension: three turning points require degree at least four. For a nonconstant polynomial, differentiation lowers the degree by one.

Corners

Why a sharp point is different

The teacher assumes an input of 7 with one-sided slopes 8 and −2. These are illustrative values, not measurements from the unlabeled sketch.

\begin{gathered}f'_-(7)=8,\quad f'_+(7)=-2\\\Longrightarrow f'(7)\text{ does not exist}\end{gathered}

The one-sided limits of the difference quotient disagree:

\lim_{x\to a^-}\frac{f(x)-f(a)}{x-a}\ne\lim_{x\to a^+}\frac{f(x)-f(a)}{x-a}

Use open circles in the derivative graph at the corner. The function itself need not have a jump.

Familiar families

Cosine and the natural exponential

f(x)=\cos x\quad\Longrightarrow\quad f'(x)=-\sin x

Cosine has horizontal tangents at its peaks and troughs. From zero to pi it falls, so its derivative is negative there. The sketch matches negative sine.

f(x)=e^x\quad\Longrightarrow\quad f'(x)=e^x

The natural exponential is always increasing. Its slopes grow to the right and flatten toward zero on the left; the derivative matches the original function.

Absolute value

Reflect the negative part

|q(x)|=\begin{cases}q(x),&q(x)\ge0,\\-q(x),&q(x)<0.\end{cases}

Reflecting the negative portion of a quadratic can create sharp corners at crossings of the axis. Compare the one-sided slopes there; do not put a zero in the derivative just because the graph has a minimum.

The teacher explicitly invents a sample quadratic and a corner location to explain this idea. They are not an exact equation-and-coordinate pair for the sketch.

Straight pieces

A line produces a constant

f(x)=mx+b\quad\Longrightarrow\quad f'(x)=m

Each straight piece has a fixed tangent slope, so its derivative is a horizontal segment. For the displayed zigzag:

f'(x)=\begin{cases}-1,&-10<x<-5,\\1,&-5<x<0,\\-1,&0<x<5,\\1,&5<x<10.\end{cases}

At corners and domain endpoints, the ordinary two-sided derivative is undefined. Do not connect across those inputs.

Vertical tangents

Continuous does not mean differentiable

f(x)=\sqrt[3]{x},\qquad f'(0)\text{ does not exist}

The cube-root graph is continuous and increasing. Near zero its tangents become arbitrarily steep; the derivative graph rises without bound on either side.

An upper semicircle has zero slope at its top, positive slopes on the left, negative slopes on the right, and unbounded slope magnitude near its endpoints.

Supplemental notation: a centered upper semicircle with $r>0$ is $f(x)=\sqrt{r^2-x^2}\quad(r>0)$ . No numerical radius is inferred from the classroom drawing.

05 · Avoid

Common mistakes

Copying the original curve.

Derivative height represents slope, not original height.

A zero at every extremum.

A smooth horizontal tangent gives zero. A sharp corner generally gives no derivative.

Confusing a root with a zero slope.

Crossing the axis does not imply a horizontal tangent.

Joining across a break.

Check corners, vertical tangents, and domain boundaries before connecting pieces.

Inventing coordinates.

An unlabeled sketch generally supports shape and signs, not exact numerical slopes.

Saying only “it is sharp.”

Explain the unequal one-sided difference quotients at a corner.

06 · Check yourself

Quick check

Supplemental questions based on the lesson. Try each before revealing the answer.

1A curve is below the axis but has positive tangent slopes. What sign does its derivative have?

Positive. Tangent slope, not the curve's height, determines derivative sign.

2A smooth curve has a horizontal tangent. Where is the derivative at that input?

On the horizontal axis: $f'(a)=0$ .

3The one-sided slopes are 8 and −2. Is the derivative their average?

No. The one-sided limits disagree, so the derivative does not exist. Averaging does not repair the limit.

4What does the derivative of a straight segment look like?

A horizontal segment whose height equals the original slope: $f(x)=mx+b\quad\Longrightarrow\quad f'(x)=m$ .

5Why is there no finite derivative of the cube-root function at zero?

The tangent is vertical. Nearby slopes become unbounded instead of approaching a finite number.

07 · Revisit

Transcript and practice

Search the corrected transcript for a term or example. Captions follow the edited video timeline.

Download corrected transcript

Class practice: use Sketching Derivatives in the Friday 9/4 eKadence activity, alongside the original supporting worksheets.