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Find f′(0) for f=(3x+1)².

Read the idea, work independently, then explain what changed.

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高二選擇性必修 第二册(A版).pdf · 5.2 · PDF 77 / printed page 72

Revisit first: Rates and tangent meaning

TOPIC 01

Derivative rules and domains

Choose power, product, quotient or chain rules from the expression structure, retaining domains.

What you will be able to explain

  • Choose power, product, quotient or chain rules from the expression structure, retaining domains.
  • Recover an unknown inner rate and audit missing derivative factors.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

Find f′(0) for f=(3x+1)².

f=(3x+1)2f=(3x+1)^2
  • Use the stated real domain and distinguish stationary candidates, attained extrema and actual concavity changes.
Skills and prerequisite lessons

Scope: JM02 · This task uses elementary probability rules or polynomial calculus named in the JAE syllabus.

  1. 2
  2. 3
  3. 9
  4. 6
Working and explanation
BUILD THE REASONING
Hint 1
Keep both the outer derivative and inner derivative.
Hint 2
Start from this intermediate relation.
f′=6(3x+1)f'=6(3x+1)
Worked solution
  1. Keep both the outer derivative and inner derivative.
  2. Establish and simplify this mathematical relation.
    f′=6(3x+1)f'=6(3x+1)
  3. Establish and simplify this mathematical relation.
    f′(0)=6f'(0)=6
  4. The inner factor is 3.
6
Checks and common pitfalls: The inner factor is 3.
Think first. Reveal a hint when the class is ready.
Focus on one question
Teacher preparation and assessment
40-minute lesson
MinutesFocusTeaching and student action
5PredictPredict the derivative of x·x before expanding.
8Build the ideaSample response: each factor contributes 1·x, so the sum is 2x.
10Guided comparisonSample response: F′(2)=4g′(4)=24, so g′(4)=6.
12Independent practice and discussionSort tasks by their outermost operation, then contrast polynomial and extension domains.
5Exit and next stepExit: (3x+1)² differentiates to 6(3x+1), and ln(3x−2) needs x>2/3.
Question sequence
  • Predict the derivative of x·x before expanding.
  • Sample response: F′(2)=4g′(4)=24, so g′(4)=6.
Board development
  • Differentiate the structure
    (uv)′=u′v+uv′; (f∘g)′=f′(g)g′(uv)'=u'v+uv';\ (f\circ g)'=f'(g)g'
Expected correct and incorrect responses
  • Correct sample: 6
  • Anticipated wrong response: 2 — Omits the inner derivative in a composition.
  • Anticipated wrong response: 6x³ — Multiplies derivatives instead of adding the two product-rule contributions.
  • Omitting the inner derivative.
  • Multiplying or dividing derivatives as if the original arithmetic rules still apply.
Follow-up questions
  • Follow-up: Keep both the outer derivative and inner derivative. Then explain why 2 fails.
  • Follow-up: First combine the powers, then differentiate. Then explain why 6x³ fails.
  • At k=0, explain why the function is constant and its derivative vanishes.
Remediation
  • Sample response: each factor contributes 1·x, so the sum is 2x.
  • Return to the first example, identify the missing condition, then solve one fresh task without viewing its solution.
Exit check
  • Exit: (3x+1)² differentiates to 6(3x+1), and ln(3x−2) needs x>2/3.
Assessment checklist and sample responses
  • Representation and conditions
    Both factors are differentiable; (uv)′=u′v+uv′.
  • Connected mathematical reasoning
    For (x²+1)(x−2), derivative 2x(x−2)+(x²+1)=3x²−4x+1.
  • Interpretation and boundary check
    ln(3x−2) needs x>2/3, not merely x≠2/3.
No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.
Curriculum and source notes ↗