How can you differentiate an inverse without its formula?
You will be able to: Find an inverse derivative using the original function’s values and 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 can you differentiate an inverse without its formula?
A temperature sensor maps an actual reading of 20 degrees to a display reading of 50 units. If its local sensitivity is 2 display units per degree, reversing the calibration gives 1/2 degree per display unit.
A useful starting point: Why are inverse-function slopes reciprocal? →
Words and symbols before equations
- Preimage
- The original input associated with a desired output.
- Inverse input
- An output of the original function.
- Reciprocal units
- Units reversed by taking 1 over a nonzero rate.
- Local inverse conditions
- A suitable one-to-one differentiable branch with nonzero derivative at the point.
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
To evaluate g′(b) for g=f⁻¹, find a such that f(a)=b. Then take 1/f′(a). You do not need a closed formula for g.
Suppose f(2)=7 and f′(2)=−4 on a valid invertible branch. Then g(7)=2 and g′(7)=−1/4.
An entry f′(7) answers a different question. Also, 1/f(2)=1/7 is a reciprocal value, not an inverse derivative.
Tables provide sufficient exact data only when the needed function and derivative entries and invertibility conditions are stated. A few distinct rows alone do not prove one-to-one behavior between them.
A worked example, step by step
A differentiable one-to-one f has f(3)=8, f′(3)=5 and f′(8)=2. Find (f⁻¹)′(8).
- Find the preimage of 8: it is 3.
- Use the original derivative at 3, not 8.
- Since f′(3)=5≠0, the local inverse formula applies.
- The answer is 1/5; the unrelated entry f′(8)=2 is unused.
Taking 1/f′(b) instead of 1/f′(f⁻¹(b)) confuses the two input systems.
Can a table of function values alone give the exact reciprocal derivative?
Compare with an explanation
Usually no. The needed original derivative must be known or determined independently.
Predict. Change one thing. Explain.
Choose a=0.5, 1 and 2. Record the inverse input a² and inverse slope 1/(2a), keeping the two columns distinct.
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 questionFor an invertible sensor calibration C, C(10)=24 and C′(10)=1.5 display units per degree. Find the inverse value and derivative at 24.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: C⁻¹(24)=10 degrees.
- 1 point: Use the preimage 10 in the derivative.
- 1 point: (C⁻¹)′(24)=1/1.5=2/3.
- 1 point: The inverse derivative has units degrees per display unit.
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 an inverse formula?
No, corresponding function and derivative values can suffice.
RECALL 2Which derivative entry matters?
The original derivative at the preimage.
RECALL 3Can table rows alone prove global invertibility?
No; behavior between and beyond the rows also matters.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How can you differentiate an inverse without its formula?
- First solve f(a)=b; then compute 1/f′(a).
- Differentiate an inverse at an original output.
Remember: Taking 1/f′(b) instead of 1/f′(f⁻¹(b)) confuses the two input systems.
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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