How do rates connect through two functions?
You will be able to: Identify inner and outer functions and multiply their local rates.
BC foundation: Unit 3 shares these differentiation objectives with AB. Track the input at each stage of a composition, preserve branch and domain conditions, and explain why your chosen derivative rule applies. Parametric and polar derivatives come later.
How do rates connect through two functions?
A square’s side grows by 2 centimeters each second. When its side is 3 centimeters, adding a little more side length changes its area faster than it did when the square was smaller.
A useful starting point: Prerequisite: product rule and derivative notation →
Words and symbols before equations
- Composition
- Applying one function to the output of another.
- Inner function
- The first operation; here side length s(t)=2t+1.
- Outer function
- The second operation; here area A(s)=s².
- Chain rule
- The local rate through a composition is the product of its two local rates.
What this picture assumes
Original mathematical model. Readouts are rounded; algebra supplies exact conclusions. s=2t+1 cm and A=s² cm². Ideal square; t≥0. The graph shows area against time, not a physical path.
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.
- At t=1 s, side=3 cm; dA/ds=6 cm²/cm and ds/dt=2 cm/s give dA/dt=12 cm²/s.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
At t=1 second, the inner output is s=3 cm. The area rule’s local rate is dA/ds=2s=6 cm² per cm at that side length.
The side changes at ds/dt=2 cm/s. Multiplying gives dA/dt=(6 cm²/cm)(2 cm/s)=12 cm²/s. The intermediate length unit cancels.
In general, if y=f(u) and u=g(x), then dy/dx=f′(g(x))g′(x). The outer derivative must be evaluated at the inner output, not automatically at x.
The chain rule requires g to be differentiable at x and f to be differentiable at g(x). This is a composition, not a product of the two function values.
A worked example, step by step
Find the area rate at t=2 s for s(t)=2t+1 cm and A=s².
- Evaluate the inner function: s(2)=5 cm.
- Differentiate the outer function: dA/ds=2s, giving 10 cm²/cm.
- Differentiate the inner function: ds/dt=2 cm/s.
- Multiply: dA/dt=20 cm²/s; this is an area rate, not the area 25 cm².
Multiplying function values does not give a derivative. Evaluate the outer derivative at the inner output.
Which factor remains constant?
Compare with an explanation
ds/dt=2 cm/s stays constant; dA/ds=2s increases as the square grows.
Predict. Change one thing. Explain.
Move t from 0 to 2. Predict how the side rate, area per side rate, and total area rate change. Explain their units.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At t=1 s, side=3 cm; dA/ds=6 cm²/cm and ds/dt=2 cm/s give dA/dt=12 cm²/s.
Original mathematical model. Readouts are rounded; algebra supplies exact conclusions. s=2t+1 cm and A=s² cm². Ideal square; t≥0. The graph shows area against time, not a physical path.
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 questionA square has side s(t)=3t+2 cm. Find its area and area rate at t=1 s, explaining the two derivative factors.
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Compare with the answer and four-point rubric
- 1 point: s(1)=5 cm and A=25 cm².
- 1 point: dA/ds=2s=10 cm²/cm.
- 1 point: ds/dt=3 cm/s.
- 1 point: dA/dt=30 cm²/s; area and area rate have different units.
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 makes a composition?
One function consumes another function’s output.
RECALL 2Where is the outer derivative evaluated?
At g(x).
RECALL 3Why multiply the rates?
The output change per intermediate unit multiplies the intermediate change per input unit.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do rates connect through two functions?
- (f∘g)′(x)=f′(g(x))g′(x).
- For s=2t+1, d(s²)/dt=2(2t+1)·2.
Remember: Multiplying function values does not give a derivative. Evaluate the outer derivative at the inner output.
Conditions: Original mathematical model. Readouts are rounded; algebra supplies exact conclusions. s=2t+1 cm and A=s² cm². Ideal square; t≥0. The graph shows area against time, not a physical path.
Refresh Kid · AP Calculus BC 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 BC 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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