Why are inverse-function slopes reciprocal?
You will be able to: Connect swapped coordinates, reflected tangents and reciprocal derivative values.
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.
Why are inverse-function slopes reciprocal?
A square with side 2 has area 4. Reversing that calculation takes area 4 back to side 2. Reversing the nearby rate also swaps its input and output roles.
A useful starting point: How do you find and classify an implicit tangent? →
Words and symbols before equations
- Inverse function
- A rule that reverses a one-to-one function’s input–output mapping.
- One-to-one
- Different allowed inputs give different outputs.
- Reflection
- A coordinate swap across the line y=x.
- Reciprocal rate
- 1 divided by a nonzero rate, with reversed units.
What this picture assumes
Original mathematical model. Readouts are rounded; algebra supplies exact conclusions. f=x² restricted to x≥0; inverse g=√x. Equal-scale 2D axes show both exact graphs. 3D folding is a coordinate construction: only 180° gives the reflected inverse; intermediate depth is not a function variable. Camera rotation does not change the fold or values.
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.
- P(1.5, 2.25) ↔ Q(2.25, 1.5). f′(1.5)=3; inverse slope at 2.25 is 0.333333; product=1. Fold=0°: original graph position.
- 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 f(x)=x² restricted to x≥0, the inverse is g(x)=√x. The point (a,a²) becomes (a²,a). This is different from taking the reciprocal 1/x².
Differentiate f(g(x))=x: f′(g(x))g′(x)=1. Therefore g′(x)=1/f′(g(x)), when the required local inverse exists and the denominator is nonzero.
At a=2, f′(2)=4 and g′(4)=1/4. The slopes are evaluated at corresponding inputs, 2 and 4, not at the same input.
The optional 3D fold rotates a graph around the line y=x by 180°. Its endpoint is the inverse reflection. Intermediate depth is a construction step, not a third variable in the function. The equal-scale 2D diagram and text contain the full mathematics.
| Feature | Inverse f⁻¹ | Reciprocal 1/f |
|---|---|---|
| Operation | Reverses a one-to-one mapping | Divides 1 by an output |
| For f=x², x≥0 | √x | 1/x² for x>0 |
| Derivative at x=4 | 1/4 | −1/32 |
A worked example, step by step
For f(x)=x² on x≥0, find (f⁻¹)′(9).
- Find the original input: f(3)=9.
- Differentiate f: f′(x)=2x.
- Evaluate f′(3)=6, which is nonzero.
- The inverse derivative at 9 is 1/6; its input is the original output.
f⁻¹ means inverse function; 1/f means reciprocal function. They usually have different derivatives.
Does the inverse slope use f′ at b?
Compare with an explanation
No. It uses f′ at a, where f(a)=b.
Predict. Change one thing. Explain.
Move a, then fold from 0° to 180°. Identify the fixed hinge y=x, the swapped point and the two tangent slopes. Explain why intermediate depth is only a construction.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
P(1.5, 2.25) ↔ Q(2.25, 1.5). f′(1.5)=3; inverse slope at 2.25 is 0.333333; product=1. Fold=0°: original graph position.
Original mathematical model. Readouts are rounded; algebra supplies exact conclusions. f=x² restricted to x≥0; inverse g=√x. Equal-scale 2D axes show both exact graphs. 3D folding is a coordinate construction: only 180° gives the reflected inverse; intermediate depth is not a function variable. Camera rotation does not change the fold or values.
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 questionUse f(x)=x², x≥0, to compare f′(1.5) and (f⁻¹)′(2.25), and explain their product.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: f(1.5)=2.25 establishes the correspondence.
- 1 point: f′(1.5)=3.
- 1 point: The inverse derivative at 2.25 is 1/3.
- 1 point: Their product is 1 because the composed mapping returns its input.
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 line reflects a graph to its inverse?
y=x, with equal coordinate scales.
RECALL 2What point corresponds to (a,b)?
(b,a).
RECALL 3What happens to derivative units?
Output per input becomes input per output.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Why are inverse-function slopes reciprocal?
- If f(a)=b and f′(a)≠0 under the local inverse conditions, (f⁻¹)′(b)=1/f′(a).
- Reflection swaps (a,b) with (b,a).
Remember: f⁻¹ means inverse function; 1/f means reciprocal function. They usually have different derivatives.
Conditions: Original mathematical model. Readouts are rounded; algebra supplies exact conclusions. f=x² restricted to x≥0; inverse g=√x. Equal-scale 2D axes show both exact graphs. 3D folding is a coordinate construction: only 180° gives the reflected inverse; intermediate depth is not a function variable. Camera rotation does not change the fold or values.
Refresh Kid · AP Calculus BC Unit 3 · Objectives FUN-3.E · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 3.3, FUN-3.E. 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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