How do you use the chain rule with tables?
You will be able to: Use the inner output to locate the correct outer derivative in data.
How do you use the chain rule with tables?
Two calibration devices are connected. The first sends input 2 to output 5. To find the combined sensitivity at input 2, you need the second device’s sensitivity at 5.
A useful starting point: How do trigonometric, exponential and logarithmic layers combine? →
Words and symbols before equations
- Preimage
- An input that maps to a specified output.
- Table entry
- A stated value at one input, not a rule between rows.
- Sensitivity
- Local output rate per unit of input.
- Exact derivative data
- Given local rates, distinct from estimates using nearby function values.
What this picture assumes
Original mathematical model. Readouts are rounded; algebra supplies exact conclusions. Known functions g(x)=2x+1 and f(u)=u². These exact rules supply derivatives; a table of function values alone would not generally do so.
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.
- Input 1 → g(a)=3 → f′(g(a))=6. Multiply by g′(a)=2 to get 12.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
If H=f∘g, first find g(a). Then read f′ at that output and g′ at a. The product is H′(a)=f′(g(a))g′(a).
Suppose g(2)=5, g′(2)=−3 and f′(5)=4. Then H′(2)=−12. An unrelated value f′(2) must not replace f′(5).
A table of function values alone generally cannot establish an exact derivative. You may estimate a slope from nearby rows, labeling it as an estimate.
The graph model uses known functions so its displayed derivatives are exact. The table-style questions instead assume the derivative entries are given and the required differentiability holds.
A worked example, step by step
Given g(1)=4, g′(1)=2, f′(1)=9 and f′(4)=−3, find (f∘g)′(1).
- The inner input is 1 and its output is g(1)=4.
- Read the outer derivative at 4: f′(4)=−3.
- Read the inner derivative at 1: g′(1)=2.
- Multiply to get −6; f′(1)=9 is not used.
Reading both derivatives from the same input row often gives the wrong result.
What if f′(g(a)) is missing?
Compare with an explanation
Without another valid way to determine it, the table is insufficient for the exact composite derivative.
Predict. Change one thing. Explain.
Move a in g(x)=2x+1, f(u)=u². Trace a → g(a) → f′(g(a)) and explain why the outer derivative’s input differs from a.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Input 1 → g(a)=3 → f′(g(a))=6. Multiply by g′(a)=2 to get 12.
Original mathematical model. Readouts are rounded; algebra supplies exact conclusions. Known functions g(x)=2x+1 and f(u)=u². These exact rules supply derivatives; a table of function values alone would not generally do so.
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 questionLet g(−1)=3, g′(−1)=−2, f(3)=7 and f′(3)=5. Find H(−1) and H′(−1) for H=f∘g.
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Compare with the answer and four-point rubric
- 1 point: The inner output is 3.
- 1 point: H(−1)=f(3)=7.
- 1 point: H′(−1)=f′(3)g′(−1).
- 1 point: H′(−1)=−10; value and rate use different entries.
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 is the first lookup?
Find g(a).
RECALL 2What is the second lookup?
Find f′ at g(a).
RECALL 3What does a negative product mean?
The composed output is locally decreasing as its original input increases.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do you use the chain rule with tables?
- H′(a)=f′(g(a))g′(a).
- Given derivatives are exact data; slopes estimated from function values are approximations.
Remember: Reading both derivatives from the same input row often gives the wrong result.
Conditions: Original mathematical model. Readouts are rounded; algebra supplies exact conclusions. Known functions g(x)=2x+1 and f(u)=u². These exact rules supply derivatives; a table of function values alone would not generally do so.
Refresh Kid · AP Calculus AB Unit 3 · Objectives FUN-3.C · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 3.1, FUN-3.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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