How do you choose the first derivative rule?
You will be able to: Read the outermost operation before selecting and combining rules.
How do you choose the first derivative rule?
Compare (x²+1)³ with x²(x+1). Both contain powers and parentheses, but the first cubes one expression and the second multiplies two expressions.
A useful starting point: How do branches and absolute values guide inverse-trig formulas? →
Words and symbols before equations
- Outermost operation
- The last operation used to form the whole expression.
- Structure
- How subexpressions are combined, independent of their appearance.
- Equivalent expression
- A different algebraic form with the same values on the stated domain.
- Procedure
- A justified sequence of differentiation steps.
What this picture assumes
Original mathematical model. Readouts are rounded; algebra supplies exact conclusions. All quantities dimensionless. Original quotient excludes x=1 even after cancellation; the hole is marked with an open circle.
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.
- Chain rule: 3(x²+1)²·2x. At x=1, value=8, derivative=24.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Read the complete expression from the outside. A sum separates into derivatives; a product starts with u′v+uv′; a quotient uses its quotient rule; an outer function wrapping an inner expression uses the chain rule.
For (x²+1)³, the outside is a cube, so chain rule comes first. For x²(x+1), multiplication is outside, so use product rule or expand first.
Simplification may reduce the work. (x²−1)/(x−1)=x+1 only for x≠1. Its derivative is 1 on that original domain, but the original function still has a hole at 1.
Topic 3.5 integrates earlier derivative objectives and selection of procedures. More than one valid method is possible; explain why your chosen method fits the expression.
A worked example, step by step
Differentiate y=(x²−1)/(x−1) and state its derivative domain.
- Record the original restriction x≠1.
- Factor the numerator to (x−1)(x+1).
- Cancel only on that domain, obtaining y=x+1.
- Thus y′=1 for x≠1; no derivative of the original function exists at x=1.
Canceling a factor simplifies calculations but does not fill a hole in the original function.
Must x²(x+1) use the product rule?
Compare with an explanation
No. Expanding to x³+x² and applying the power/sum rules is also valid.
Predict. Change one thing. Explain.
Switch among a power, product and simplified quotient. Before revealing the readout, name the first rule and identify any excluded inputs.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Chain rule: 3(x²+1)²·2x. At x=1, value=8, derivative=24.
Original mathematical model. Readouts are rounded; algebra supplies exact conclusions. All quantities dimensionless. Original quotient excludes x=1 even after cancellation; the hole is marked with an open circle.
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 questionDifferentiate x²(x+1) by expanding and by the product rule, and compare the results.
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Compare with the answer and four-point rubric
- 1 point: Expansion gives x³+x².
- 1 point: Its derivative is 3x²+2x.
- 1 point: The product rule gives 2x(x+1)+x².
- 1 point: Simplification also gives 3x²+2x, confirming agreement for all real x.
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 should you inspect first?
The outermost operation of the whole expression.
RECALL 2Can two methods both be valid?
Yes, if their rules and algebra apply on the original domain.
RECALL 3What survives cancellation?
Original excluded inputs remain excluded unless you explicitly define a different extended function.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do you choose the first derivative rule?
- Identify the outermost operation first.
- Carry the original domain through algebraic simplification.
Remember: Canceling a factor simplifies calculations but does not fill a hole in the original function.
Conditions: Original mathematical model. Readouts are rounded; algebra supplies exact conclusions. All quantities dimensionless. Original quotient excludes x=1 even after cancellation; the hole is marked with an open circle.
Refresh Kid · AP Calculus AB Unit 3 · Objectives FUN-3.A–E; Mathematical Practice 1.C · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 3.5, FUN-3.A–E; Mathematical Practice 1.C. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 3 has six official topics. Topic 3.5 integrates existing derivative objectives and Mathematical Practice 1.C; it does not introduce a separate new objective. Focused lesson titles and questions are original Refresh Kid teaching material.
Chain rule factors are evaluated at their correct nested inputs. Implicit derivatives refer to local branches and retain denominator conditions. Inverse derivatives require the corresponding preimage and a nonzero original derivative under the local inverse conditions. Trigonometric inputs and returned angles use radians; inverse branch conventions and domains are stated. Related rates and L’Hôpital’s rule remain in later units. Higher derivatives are repeated differentiation, not powers.
The Organic Chemistry Tutor video creator and relevant descriptions were checked (the implicit video link is labeled descriptively); 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 3.6, 3.7 and 3.8 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 folds a graph 180 degrees around y=x to construct its inverse reflection. Intermediate depth is a geometric construction, not an extra function variable. Camera rotation only changes the view. Equal-scale labeled 2D graphs and text provide the complete explanation.
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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