The Product Rule
Account for both changing factors in a product, whether values come from formulas or data.
College Board Topic 2.8 · CED p. 62 (PDF p. 67) · CED effective Fall 2020; project course outline 2026–27.
When both factors vary, each contributes to the rate. The product rule is (fg)′=f′g+fg′ when f and g are differentiable. Think “change the first, keep the second; keep the first, change the second; add.” Each term accounts for one source of variation.
For x²sin x, the derivative is 2x sin x+x²cos x. Multiplying 2x by cos x misses the two contributions. A quick check with x·x makes the error clear: x² differentiates to 2x, while the product of derivatives would be 1.
The same rule works with data. If f(2)=3, f′(2)=−1, g(2)=4 and g′(2)=5, then (fg)′(2)=−1·4+3·5=11. Make sure the four values all use the same input.
Let's work through it
- Let f=x² and g=sin x. Then f′=2x and g′=cos x. Reason: the factors are familiar differentiable functions.
- Apply f′g+fg′: 2x sin x+x²cos x. Reason: both factors vary.
- Keep the two terms or factor x if useful. Reason: algebraic simplification may follow, but it does not change the differentiation rule.
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 xeˣ.
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 xeˣ.
Worked answer and reason
eˣ+xeˣ.
Both the x factor and the exponential factor contribute.
Retrieval 2
If f(1)=2, f′(1)=3, g(1)=4, g′(1)=−1, find (fg)′(1).
Worked answer and reason
10.
The product rule gives 3·4+2·(−1).
Catch the tempting mistake
- The derivative of a product is f′g′.
- The correct formula adds f′g and fg′; x·x supplies an immediate counterexample.
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 23 · No calculator · analytic
- Unit 2 problem 24 · No calculator · numerical
- Unit 2 problem 25 · No calculator · verbal
For the teacher
- Use a changing rectangle to interpret the two product-rule contributions.
- Mix formula, table and verbal product contexts.
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.