How do you differentiate a product without formulas?
You will be able to: Combine supplied values and derivatives at a shared input.
How do you differentiate a product without formulas?
A model gives you two factors’ values and rates at one instant but no complete equations. You can still calculate the product’s rate at that instant.
A useful starting point: Why does a changing product need two terms? →
Words and symbols before equations
- Pointwise data
- Values supplied at one specified input.
- u(a)
- The value of the first factor at a.
- u′(a)
- Its local rate at a.
- Weighted contribution
- A rate multiplied by the other factor’s current value.
What this picture assumes
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. All four values are supplied at the same input; differentiability is assumed. Local data do not specify the full functions.
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.
- The two weighted contributions 8 and -3 sum to 5. Supplied values assume differentiability; they do not specify entire graphs.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
The product rule is local: (uv)′(a)=u′(a)v(a)+u(a)v′(a). Full formulas are unnecessary if all four pointwise values are known and differentiability is given.
With u=3, u′=2, v=4 and v′=−1 at a, the contributions are 2·4=8 and 3·(−1)=−3, for a net rate 5.
A falling factor can subtract from the product rate without making the net rate negative; the other contribution may be larger.
Use values at the same input. Mixing u(a) with v(b) does not evaluate a derivative of their product at either point.
A worked example, step by step
At a=1, u(1)=2, u′(1)=−3, v(1)=5, v′(1)=4. Find (uv)′(1).
- Write the local rule u′(1)v(1)+u(1)v′(1).
- First contribution=−3·5=−15.
- Second contribution=2·4=8.
- Add to obtain −7; the product is decreasing at that instant.
A function value and its derivative are different entries. Do not swap them in a table.
Can the product rate be zero when neither factor rate is zero?
Compare with an explanation
Yes. The two weighted contributions can cancel.
Predict. Change one thing. Explain.
Change the supplied values and rates. Predict which contribution is positive and find a case where the two contributions exactly cancel.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
The two weighted contributions 8 and -3 sum to 5. Supplied values assume differentiability; they do not specify entire graphs.
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. All four values are supplied at the same input; differentiability is assumed. Local data do not specify the full functions.
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 questionAt a point u=4,u′=2,v=3. Find v′ if the product’s derivative is zero, and explain the cancellation.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Write 0=u′v+uv′.
- 1 point: Substitute 0=2·3+4v′.
- 1 point: Solve v′=−6/4=−1.5.
- 1 point: The contributions +6 and −6 cancel; neither factor needs to have zero rate.
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 1Do you need full formulas?
No, supplied local values and rates suffice.
RECALL 2Must all values use the same input?
Yes.
RECALL 3Can nonzero contributions sum to zero?
Yes; rates can offset each other.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do you differentiate a product without formulas?
- (uv)′(a)=u′(a)v(a)+u(a)v′(a).
Remember: A function value and its derivative are different entries. Do not swap them in a table.
Conditions: Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. All four values are supplied at the same input; differentiability is assumed. Local data do not specify the full functions.
Refresh Kid · AP Calculus AB Unit 2 · Objectives FUN-3.B · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 2.8, 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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