How do local slopes become a new function?
You will be able to: Connect values of a function to the corresponding values of its derivative.
How do local slopes become a new function?
On a gently curved ramp graph y=x², the tangent is level at x=0, slopes upward at x=1 and more steeply upward at x=2. We can collect those slopes as outputs of a second rule.
A useful starting point: How does a limit define an instantaneous rate? →
Words and symbols before equations
- Derivative function
- The rule assigning f′(x) to each input where the derivative exists.
- Prime notation
- f′ names the derivative, not a new power of f.
- dy/dx
- Another notation for the derivative when y=f(x); introduced here as one symbol.
- Domain of f′
- Inputs where the derivative exists as a finite real number.
What this picture assumes
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. f(x)=x² and f′(x)=2x, dimensionless coordinates. The second graph height records the first graph slope at the same input. Optional 3D depth separates panels, not variables.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- a=1; f(a)=1; f′(a)=2. Tangent: y−1=2(x−(1)). Blue: f=x²; orange: tangent. The second blue graph is f′=2x. All coordinates are dimensionless; graph heights and rates have distinct roles.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Keep x variable in [(x+h)²−x²]/h. Expanding and cancelling h≠0 yields 2x+h; taking its limit gives f′(x)=2x.
The input x stays the same in the two graphs. A point (a,a²) on f corresponds to (a,2a) on f′ because the second height records the first graph’s slope.
At x=−1, f is positive with height 1, yet its derivative is negative with value −2. Function sign and slope sign answer different questions.
The optional 3D view separates the two graph planes to inspect this correspondence. Depth distinguishes the two panels only; this is not a multivariable function. Complete paired 2D graphs carry all coordinates.
| Feature | f(a) | f′(a) |
|---|---|---|
| Meaning | Output at the input a | Local rate at the same input a |
| Graph role | Height on f | Tangent slope on f; height on f′ |
| Units | Output units | Output units per input unit |
A worked example, step by step
For f(x)=x², find f(−2), f′(x), and f′(−2).
- The original output is f(−2)=4.
- The general difference quotient is 2x+h for h≠0.
- Taking h→0 gives f′(x)=2x.
- Thus f′(−2)=−4: the function is above zero but locally decreasing.
f′ is neither 1/f nor the square of f. A positive function value does not force a positive derivative.
At x=0 why is the derivative graph on the horizontal axis?
Compare with an explanation
The tangent to x² is horizontal there, so its slope is zero.
Predict. Change one thing. Explain.
Move the shared input from −2 to 2. Compare the blue function height, orange tangent slope and derivative height. Optionally rotate the paired planes and identify their shared input.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
a=1; f(a)=1; f′(a)=2. Tangent: y−1=2(x−(1)). Blue: f=x²; orange: tangent. The second blue graph is f′=2x. All coordinates are dimensionless; graph heights and rates have distinct roles.
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. f(x)=x² and f′(x)=2x, dimensionless coordinates. The second graph height records the first graph slope at the same input. Optional 3D depth separates panels, not variables.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using a difference quotient, tangent slope, or derivative rule with its domain conditions. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionFor f(x)=x², describe corresponding points on f and f′ at x=−1 and x=1 and explain their slope signs.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: f(−1)=1 and f(1)=1.
- 1 point: f′(x)=2x.
- 1 point: Derivative points are (−1,−2) and (1,2).
- 1 point: Equal original heights can have opposite tangent slopes.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What does a height on f′ mean?
The slope of f at the same input.
RECALL 2What is dy/dx?
Derivative notation for y with respect to x.
RECALL 3Does 3D depth add an input?
No. It separates graph panels for inspection only.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do local slopes become a new function?
- For f(x)=x², f′(x)=2x.
- y′, f′(x), and dy/dx describe the same derivative when y=f(x).
Remember: f′ is neither 1/f nor the square of f. A positive function value does not force a positive derivative.
Conditions: Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. f(x)=x² and f′(x)=2x, dimensionless coordinates. The second graph height records the first graph slope at the same input. Optional 3D depth separates panels, not variables.
Refresh Kid · AP Calculus AB Unit 2 · Objectives CHA-2.B, CHA-2.C · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 2.2, CHA-2.B, CHA-2.C. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 2 has 10 official topics. Focused lesson titles and questions are original Refresh Kid teaching material.
Derivative definitions require finite two-sided limits at interior inputs. Graphs and finite tables supply evidence and estimates, not universal proofs. Continuity is necessary but insufficient for differentiability. Rules retain their original domain restrictions, and trigonometric inputs use radians. Chain-rule compositions, implicit differentiation, inverse-function differentiation and L’Hôpital’s rule are deferred to later units. Piecewise joins are checked for continuity before their side slopes are compared.
The Organic Chemistry Tutor video titles, creators and descriptions were checked; full videos were not reviewed. Khan Academy’s unit destination was checked; its JavaScript lesson content was not fully readable by the research tool. OpenStax Sections 3.1, 3.3 and 3.5 were consulted for conceptual cross-checking. No creator scripts, questions, diagrams or artwork were copied. Refresh Kid is not endorsed by these providers.
GitHub’s 3D website collection and its camera-control example informed optional spatial inspection. Graph geometry is generated from the stated original equations; self-hosted Three.js retains its MIT license. The optional 3D model uses separate planes for f and f′. Depth distinguishes panels, not an additional variable; camera rotation does not change their values. Use the complete labeled 2D graphs and text to read precise coordinates and slopes.
Independent teacher review and observation of students remain pending. Checks do not certify scientific accuracy, accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus AB questions and scoring guides. The archive spans multiple units; no entire exam question is assigned to this lesson.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
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