Kenneth ChengMathematical modeling
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Find f′(1) using the product rule.

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

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TOPIC 01

Derivatives and their applications: mixed review

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

  • Connect the chapter skills without relying on the order of the exercises.

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′(1) using the product rule.

f(x)=x2(x+21)f(x)=x^2(x+21)
  • Quotient denominators stay nonzero; logarithms have positive real arguments and trigonometric derivatives use radians.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Differentiate the specified function and substitute only after differentiating.
Hint 2
Use this intermediate relation.
f′=2x(x+t)+x2f\prime=2x(x+t)+x^2
Worked solution

  1. Differentiate the specified function and substitute only after differentiating.

  2. Apply the stated relation and retain its conditions.

    f′(1)=2(1+21)+1=45f'(1)=2(1+21)+1=45

  3. Both factors contribute to the derivative.

The requested value is 45.

Checks and common pitfalls: Both factors contribute to the derivative.

Think first. Reveal a hint when the class is ready.

Focus on one question

Teacher preparation and assessment

Question sequence

  • Ask students to name the relevant condition before calculating.

Board plan

  • Compare valid methods and annotate their conditions.

Anticipated thinking

  • A correct final value may still hide a missing assumption.

Assessment checklist

  • Check the method, conditions, reasoning and interpretation separately.

No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

Curriculum and source notes ↗