The Quotient Rule
Choose a quotient rule or equivalent rewrite and preserve the original denominator restrictions.
College Board Topic 2.9 · CED p. 63 (PDF p. 68) · CED effective Fall 2020; project course outline 2026–27.
For a differentiable numerator f and denominator g with g(x)≠0, (f/g)′=(f′g−fg′)/g². The numerator order matters. “Bottom times derivative of top minus top times derivative of bottom” is a useful spoken check.
For (x²+1)/(x−1), the derivative is [2x(x−1)−(x²+1)]/(x−1)²=(x²−2x−1)/(x−1)², with x≠1. Keep the denominator squared and retain the original domain. The rule does not permit evaluating at a zero denominator.
Sometimes rewriting is cleaner: (x²+1)/x=x+1/x for x≠0, so its derivative is 1−1/x². Choose a form that makes the reasoning easy, and check that simplification has not silently added inputs.
Let's work through it
- Label f=x²+1, g=x−1, f′=2x, g′=1. Reason: naming the roles prevents a sign reversal.
- Form [2x(x−1)−(x²+1)]/(x−1)². Reason: apply the quotient rule in the correct order.
- Simplify to (x²−2x−1)/(x−1)², x≠1. Reason: combine terms while keeping the domain restriction.
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 (x+1)/(x+2).
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 (x+1)/(x+2).
Worked answer and reason
1/(x+2)² for x≠−2.
The numerator is (x+2)−(x+1)=1.
Retrieval 2
Why is (f/g)′ not generally f′/g′?
Worked answer and reason
Both numerator and denominator variation affect the rate.
For x/x=1 away from 0, the derivative is 0 rather than 1/1.
Catch the tempting mistake
- Reversing the numerator order makes no difference.
- It changes the derivative sign; order is part of the rule.
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 2 problem 26 · No calculator · analytic
- Unit 2 problem 27 · No calculator · numerical
- Unit 2 problem 28 · No calculator · analytic
For the teacher
- Have students verify one quotient using an equivalent simpler expression.
- Give a tabular quotient with a nonzero denominator and a separate invalid zero-denominator case.
Glossary and deeper reading · 2 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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Differentiability
Differentiability means that a function has a finite derivative at the point under discussion. Differentiability at a point implies continuity there, but continuity alone does not imply differentiability. For example, |x| is continuous at 0 but its left and right tangent slopes differ, so it is not differentiable there. Corners, cusps, vertical tangents, and discontinuities can prevent a finite derivative.
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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.