Implicit Differentiation
Differentiate an implicit relation locally, checking the point and any division used to find its slope.
College Board Topic 3.2 · CED p. 71 (PDF p. 76) · CED effective Fall 2020; project course outline 2026–27.
Some curves are easiest to describe by a relationship between x and y. Treat y as a local function of x when differentiating an equation. Every derivative of a y-expression needs the chain rule: d(y²)/dx=2y y′. Differentiate both sides, collect the y′ terms, then solve for y′ where that division is valid.
For x²+y²=25, differentiation gives 2x+2y y′=0, so y′=−x/y when y≠0. At (3,4) the slope is −3/4. The equation describes two branches, so a supplied point identifies which local branch you mean. At y=0, the expression cannot give a finite slope; the circle has a vertical tangent at (±5,0).
Products involving x and y need both the product and chain rules: d(xy)/dx=y+x y′. Check that the requested point actually lies on the curve before calculating a slope. An implicit derivative describes the local rate along a curve, rather than a free choice of unrelated x and y values.
Let's work through it
- At (3,4) on x²+y²=25, differentiate to 2x+2y y′=0. Reason: y depends on x, so the chain rule contributes y′.
- Solve y′=−x/y. Reason: here y=4≠0 so division is valid.
- The slope is −3/4, giving tangent y−4=−(3/4)(x−3). Reason: substitute the actual curve point into the local rate.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
Differentiate xy with respect to x when y depends on x.
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Differentiate xy with respect to x when y depends on x.
Worked answer and reason
y+x y′.
Use the product rule and d(y)/dx=y′.
Retrieval 2
For x²+y²=9 at (0,3), what is the slope?
Worked answer and reason
0.
The implicit slope −x/y is 0 and the denominator is nonzero.
Catch the tempting mistake
- The derivative of y² with respect to x is 2y.
- It is 2y y′ because y varies with x.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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Put it to work
- Unit 3 problem 8 · No calculator · analytic
- Unit 3 problem 9 · No calculator · analytic
- Unit 3 problem 10 · No calculator · analytic
- Unit 3 problem 11 · No calculator · analytic
- Unit 3 problem 12 · No calculator · graphical
- Unit 3 problem 13 · Graphing calculator required · multiple
- Unit 3 problem 30 · No calculator · analytic
For the teacher
- Ask students to circle every occurrence of y before differentiating.
- Check curve membership and denominator conditions in every point-slope task.
Glossary and deeper reading · 3 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.
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Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.
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Vertical tangent
A vertical tangent is a tangent line parallel to the vertical axis, with no finite slope. A graph can have a vertical tangent even though its derivative does not exist as a finite real number at that input. For example, y=∛x has the vertical tangent x=0 at the origin as its nearby slopes tend to positive infinity. This case is separate from the finite-slope tangent approximation available at a differentiable point.