Selecting Procedures for Calculating Derivatives
Choose and combine rules from the outer structure, rather than treating a mixed expression as one memorized pattern.
College Board Topic 3.5 · CED p. 74 (PDF p. 79) · CED effective Fall 2020; project course outline 2026–27.
Read the outer structure before choosing a rule. A sum invites term-by-term differentiation; a product requires two contributions; a quotient may use the quotient rule or a helpful rewrite; a composition requires an inside-rate factor. An equation relating x and y calls for implicit differentiation. An inverse-rate problem starts by finding the corresponding original input.
Several rules may work together. In x²sin(3x), the outer structure is a product, and one factor is a composition. Differentiate the product first, then use the chain rule on sin(3x). In ln[(x+1)/(x−1)] for x>1, a logarithm identity gives ln(x+1)−ln(x−1), so two chain-rule terms are clearer than nested quotient work.
Check domains and special points before applying a formula. A simplified expression can make algebra easier but cannot automatically add a removed input. A table or graph may supply values rather than a formula; the same rule structure still applies. After calculation, check the sign or a numerical slope when useful.
Let's work through it
- For y=x²sin(3x), identify a product of x² and sin(3x). Reason: multiplication is the outer operation.
- Use the product rule: y′=2x sin(3x)+x²·[3cos(3x)]. Reason: the second factor requires the chain rule.
- Simplify to 2x sin(3x)+3x²cos(3x). Reason: algebra can follow once all rate contributions are present.
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
Which rules are needed for x·ln(1+x²)?
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
Which rules are needed for x·ln(1+x²)?
Worked answer and reason
Product rule and chain rule, with the logarithm rule.
Both factors vary and the logarithm contains an inside function.
Retrieval 2
Which method fits x²+y²=4 at a supplied curve point?
Worked answer and reason
Implicit differentiation, then point evaluation.
The relation has not isolated a single global y-function.
Catch the tempting mistake
- One expression always needs exactly one derivative rule.
- Products, quotients and compositions can be nested, so several justified rules may be needed.
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 24 · No calculator · analytic
- Unit 3 problem 25 · No calculator · analytic
- Unit 3 problem 26 · No calculator · analytic
For the teacher
- Use a rule-choice card sort before completing full derivatives.
- Ask students to compare two valid methods and explain which is clearer.
Glossary and deeper reading · 5 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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Quotient rule
The quotient rule differentiates a quotient where the denominator is nonzero and both functions are differentiable. It gives (f/g)′=(f′g−fg′)/g². For example, differentiating x/(x+1) produces 1/(x+1)² for x≠−1. The order of the subtraction and the denominator square both matter.
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Implicit differentiation
Implicit differentiation finds a derivative from an equation relating variables without first solving explicitly for one variable. When y depends on x, differentiating a term containing y requires the chain rule and a factor of dy/dx. For example, x²+y²=1 gives 2x+2y(dy/dx)=0. Solving that relation for dy/dx gives −x/y only where y≠0 and the relation supports the local function being considered.
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Chain rule
The chain rule differentiates a composition by multiplying the outer derivative at the inner output by the inner derivative. For differentiable functions, (f(g(x)))′=f′(g(x))g′(x). For example, the derivative of sin(x²) is cos(x²)·2x. Nested compositions require a factor for each successive layer.
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Logarithm
A logarithm gives the exponent required to obtain a number from a specified base. For real logarithms, the argument is positive and the base is positive and different from 1. For example, log₂8=3 because 2³=8, while ln uses the base e. Logarithm rules convert products into sums and powers into constant multiples, subject to the domain restrictions.
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Domain
The domain of a function is the set of inputs for which the function is defined. An expression and its context can both restrict the allowed inputs. For example, the real function f(x)=1/(x−2) has domain all real numbers except 2. For a model of elapsed time, a sensible domain may also require t≥0.