When does simplifying help, and what stays excluded?
You will be able to: Choose an efficient derivative method while retaining original domain exclusions.
When does simplifying help, and what stays excluded?
Cancelling a shared factor can make a ratio easy to study, but it does not repair an input where the original formula divided by zero.
A useful starting point: Why does a quotient’s derivative subtract? →
Words and symbols before equations
- Equivalent on a domain
- Two expressions agree at every input allowed by the original expression.
- Removable hole
- A missing function value despite a finite nearby limit.
- Cancellation
- Removing a shared nonzero factor in numerator and denominator.
- Restricted derivative
- A derivative formula together with the inputs where it is valid.
What this picture assumes
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. Original q=(x²−1)/(x−1), x≠1. The extension is separately defined. Hollow points remain excluded; a formula does not fill them.
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.
- Original quotient is undefined at 1. Derivative=1 for x≠1; derivative at 1 is undefined. Both open circles retain that exclusion.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
For q(x)=(x²−1)/(x−1), factor the numerator and cancel only when x≠1. The result is x+1 on that restricted domain.
Therefore q′(x)=1 for x≠1. At x=1 the original function is undefined, so its derivative is also undefined there.
The quotient rule gives [2x(x−1)−(x²−1)]/(x−1)². Its numerator is (x−1)², producing the same restricted derivative.
A different function that explicitly fills q(1)=2 would be x+1 everywhere and differentiable at 1. That is a change to the function, not merely simplification.
A worked example, step by step
Differentiate r(x)=(x²−4)/(x−2), retaining the original domain.
- The original restriction is x≠2.
- Factor x²−4=(x−2)(x+2).
- For allowed x, r=x+2, so r′=1.
- At x=2 neither r nor its derivative is defined; keep the hole.
Simplification never automatically restores an input excluded by the original function.
Does the derivative formula 1 prove the original quotient is differentiable at the hole?
Compare with an explanation
No. Formula and domain belong together; the original function has no point value there.
Predict. Change one thing. Explain.
Toggle between the original ratio and its explicitly filled extension. Explain why the line looks the same nearby but the derivative at the excluded input changes status.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Original quotient is undefined at 1. Derivative=1 for x≠1; derivative at 1 is undefined. Both open circles retain that exclusion.
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. Original q=(x²−1)/(x−1), x≠1. The extension is separately defined. Hollow points remain excluded; a formula does not fill them.
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 r(x)=(x²−16)/(x−4), state r′ on its domain and the value needed to extend r differentiably to x=4.
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Compare with the answer and four-point rubric
- 1 point: The original domain excludes 4.
- 1 point: For x≠4, r=x+4.
- 1 point: Hence r′=1 for x≠4, with no original derivative at 4.
- 1 point: Define r(4)=8 to create the line x+4 everywhere; the extension then has derivative 1 at 4.
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 1Does cancellation fill holes?
No.
RECALL 2Can a derivative exist where f is undefined?
No.
RECALL 3How can a removable hole be repaired?
Define the point value separately to equal the nearby limit; then recheck differentiability.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
When does simplifying help, and what stays excluded?
- Differentiate a simpler equivalent expression only on the original domain.
- A hole in f is also excluded from the domain of f′.
Remember: Simplification never automatically restores an input excluded by the original function.
Conditions: Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. Original q=(x²−1)/(x−1), x≠1. The extension is separately defined. Hollow points remain excluded; a formula does not fill them.
Refresh Kid · AP Calculus AB Unit 2 · Objectives FUN-3.B · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 2.9, FUN-3.B. 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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