Why does differentiating y² produce 2y y′?
You will be able to: Differentiate a relation while treating y as a local function of x.
Why does differentiating y² produce 2y y′?
A point moves along the upper half of a circle of radius 5. At (3,4), increasing x a little forces y to decrease if the point is to stay on the circle.
A useful starting point: How do you use the chain rule with tables? →
Words and symbols before equations
- Implicit relation
- An equation linking x and y without necessarily solving for y.
- Local branch
- A nearby part of the relation expressible as y(x).
- y′
- dy/dx, the local change of y per change of x.
- Differentiate with respect to x
- Measure every changing quantity against x.
What this picture assumes
Original mathematical model. Readouts are rounded; algebra supplies exact conclusions. x²+y²=25. Two separate branches, radius 5 coordinate units, equal scales. Controls exclude vertical endpoints x=±5; y′ is finite at each selected point.
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.
- Point (3, 4): y′=−x/y=-0.75; y″=−25/y³=-0.390625. The point and branch determine the finite slope.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
In x²+y²=25, y depends on x along a chosen branch. Therefore d(y²)/dx=2y·dy/dx by the chain rule, while d(x²)/dx=2x.
Differentiating both sides gives 2x+2yy′=0. Solving yields y′=−x/y when y≠0.
At (3,4), the slope is −3/4. At (3,−4), the slope is +3/4: a relation can have different local branches at the same x.
The diagram shows both branches, but a derivative value belongs to a particular point and local branch. At y=0 this solved formula is undefined; endpoint geometry needs separate analysis.
A worked example, step by step
Find the slope of x²+y²=25 at (−3,4).
- Verify the point: 9+16=25.
- Differentiate: 2x+2yy′=0.
- Isolate y′=−x/y, valid for y≠0.
- At (−3,4), y′=3/4; the upper branch rises as x increases there.
Treating y as a constant or replacing d(y²)/dx with 2y omits its dependence on x.
Is y′ always one number for an entire circle?
Compare with an explanation
No. It depends on the point and chosen local branch; the whole circle is not one function y(x).
Predict. Change one thing. Explain.
Move x and switch branches. Predict the slope signs before checking; explain why the same x can produce two different slopes.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Point (3, 4): y′=−x/y=-0.75; y″=−25/y³=-0.390625. The point and branch determine the finite slope.
Original mathematical model. Readouts are rounded; algebra supplies exact conclusions. x²+y²=25. Two separate branches, radius 5 coordinate units, equal scales. Controls exclude vertical endpoints x=±5; y′ is finite at each selected point.
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 x²+y²=25, find y′ at (4,−3) and explain its sign.
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Compare with the answer and four-point rubric
- 1 point: The point satisfies 16+9=25.
- 1 point: Differentiation gives 2x+2yy′=0.
- 1 point: Substitute to obtain y′=−4/(−3)=4/3.
- 1 point: The lower branch increases as x increases near this point.
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 1Why is there a y′ factor?
y is itself a function of x on the local branch.
RECALL 2Must you solve for y first?
No; implicit differentiation can find its local derivative directly.
RECALL 3Why name the point?
The same x can occur on more than one branch.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Why does differentiating y² produce 2y y′?
- d[y(x)²]/dx=2y y′.
- For x²+y²=R², y′=−x/y when y≠0.
Remember: Treating y as a constant or replacing d(y²)/dx with 2y omits its dependence on x.
Conditions: Original mathematical model. Readouts are rounded; algebra supplies exact conclusions. x²+y²=25. Two separate branches, radius 5 coordinate units, equal scales. Controls exclude vertical endpoints x=±5; y′ is finite at each selected point.
Refresh Kid · AP Calculus AB Unit 3 · Objectives FUN-3.D · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 3.2, FUN-3.D. 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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