Kenneth ChengMathematical modeling
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Derivative operations

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

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

Revisit first: Derivative: concept and meaning

TOPIC 01

Derivative operations

Apply power, product, quotient and chain rules with elementary derivatives.

What you will be able to explain

  • Apply power, product, quotient and chain rules with elementary derivatives.
  • Justify the method and check the conditions in a new situation.

Defining relation

Apply power, product, quotient and chain rules with elementary derivatives.

(uv)′=u′v+uv′;[f(g)]′=f′(g)g′(uv)\prime=u\prime v+uv\prime;\quad [f(g)]\prime=f\prime(g)g\prime

Conditions

Quotient denominators stay nonzero; logarithms have positive real arguments and trigonometric derivatives use radians.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Which rule is needed before substituting a point into a composite function?

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

y=ax²; tangent slope at x=t is 2; signed integral from 0 to t is 0.3333. Signed integral can be negative.

y=ax²; tangent slope at x=t is 2; signed integral from 0 to t is 0.3333. Signed integral can be negative.

Explain: Calculate two valid cases and explain the change using the defining relation.

Transfer: Compare a correct chain rule with a calculation missing the inner derivative.

Try it. Leave your reasoning visible.

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

01 / Foundation#Worked example

Find f′(1).

f(x)=x3+2xf(x)=x^3+2x
  • 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′(x)=3x2+tf\prime(x)=3x^2+t
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    f′(1)=3+2=5f'(1)=3+2=5

  3. Differentiate term by term.

The requested value is 5.

Checks and common pitfalls: Differentiate term by term.

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

02 / Standard#Worked example

Find f′(0).

f(x)=e3xf(x)=e^{3x}
  • 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′(x)=tetxf\prime(x)=te^{tx}
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    f′(0)=3e0=3f'(0)=3e^0=3

  3. The natural exponential reproduces itself under differentiation.

The requested value is 3.

Checks and common pitfalls: The natural exponential reproduces itself under differentiation.

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

03 / Transfer#Worked example

Find f′(0), using radians.

f(x)=sin⁡(4x)+cos⁡xf(x)=\sin(4x)+\cos x
  • 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′(x)=tcos⁡(tx)−sin⁡xf\prime(x)=t\cos(tx)-\sin x
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    f′(0)=4⋅1−0=4f'(0)=4\cdot1-0=4

  3. A radian argument is essential for the familiar sine derivative formula.

The requested value is 4.

Checks and common pitfalls: A radian argument is essential for the familiar sine derivative formula.

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

04 / Foundation#Your turn

Find f′(1).

f(x)=x3+5xf(x)=x^3+5x
  • 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′(x)=3x2+tf\prime(x)=3x^2+t
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    f′(1)=3+5=8f'(1)=3+5=8

  3. Differentiate term by term.

The requested value is 8.

Checks and common pitfalls: Differentiate term by term.

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

05 / Foundation#Your turn

Find f′(1) using the product rule.

f(x)=x2(x+6)f(x)=x^2(x+6)
  • 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+6)+1=15f'(1)=2(1+6)+1=15

  3. Both factors contribute to the derivative.

The requested value is 15.

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

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

06 / Foundation#Your turn

Find f′(0).

f(x)=x+7x+1f(x)=\frac{x+7}{x+1}
  • 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′=[(x+1)−(x+t)]/(x+1)2f\prime=[(x+1)-(x+t)]/(x+1)^2
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    f′(x)=(1−7)/(x+1)2f'(x)=(1-7)/(x+1)^2

  3. Apply the stated relation and retain its conditions.

    f′(0)=−6f'(0)=-6

  4. The domain excludes x=−1, and zero is admissible.

The requested value is -6.

Checks and common pitfalls: The domain excludes x=−1, and zero is admissible.

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

07 / Foundation#Your turn

Find f′(0) using the chain rule.

f(x)=(2x+8)3f(x)=(2x+8)^3
  • 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′=3(2x+t)2⋅2f\prime=3(2x+t)^2\cdot2
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    f′(0)=6(8)2=384f'(0)=6(8)^2=384

  3. The inner derivative contributes a factor of two.

The requested value is 384.

Checks and common pitfalls: The inner derivative contributes a factor of two.

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

08 / Standard#Your turn

Find f′(0) using the chain rule.

f(x)=(2x+9)3f(x)=(2x+9)^3
  • 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′=3(2x+t)2⋅2f\prime=3(2x+t)^2\cdot2
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    f′(0)=6(9)2=486f'(0)=6(9)^2=486

  3. The inner derivative contributes a factor of two.

The requested value is 486.

Checks and common pitfalls: The inner derivative contributes a factor of two.

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

09 / Standard#Your turn

Find f′(0).

f(x)=e10xf(x)=e^{10x}
  • 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′(x)=tetxf\prime(x)=te^{tx}
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    f′(0)=10e0=10f'(0)=10e^0=10

  3. The natural exponential reproduces itself under differentiation.

The requested value is 10.

Checks and common pitfalls: The natural exponential reproduces itself under differentiation.

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

10 / Standard#Your turn

Find f′(1).

f(x)=ln⁡(11x)f(x)=\ln(11x)
  • 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′(x)=t/(tx)=1/xf\prime(x)=t/(tx)=1/x
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    f′(1)=1f\prime(1)=1

  3. The scale inside the logarithm cancels, but the real domain remains x>0.

The requested value is 1.

Checks and common pitfalls: The scale inside the logarithm cancels, but the real domain remains x>0.

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

11 / Standard#Your turn

Find f′(0), using radians.

f(x)=sin⁡(12x)+cos⁡xf(x)=\sin(12x)+\cos x
  • 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′(x)=tcos⁡(tx)−sin⁡xf\prime(x)=t\cos(tx)-\sin x
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    f′(0)=12⋅1−0=12f'(0)=12\cdot1-0=12

  3. A radian argument is essential for the familiar sine derivative formula.

The requested value is 12.

Checks and common pitfalls: A radian argument is essential for the familiar sine derivative formula.

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

12 / Transfer#Your turn

Find f′(1).

f(x)=ln⁡(13x)f(x)=\ln(13x)
  • 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′(x)=t/(tx)=1/xf\prime(x)=t/(tx)=1/x
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    f′(1)=1f\prime(1)=1

  3. The scale inside the logarithm cancels, but the real domain remains x>0.

The requested value is 1.

Checks and common pitfalls: The scale inside the logarithm cancels, but the real domain remains x>0.

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

13 / Transfer#Your turn

Find f′(0), using radians.

f(x)=sin⁡(14x)+cos⁡xf(x)=\sin(14x)+\cos x
  • 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′(x)=tcos⁡(tx)−sin⁡xf\prime(x)=t\cos(tx)-\sin x
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    f′(0)=14⋅1−0=14f'(0)=14\cdot1-0=14

  3. A radian argument is essential for the familiar sine derivative formula.

The requested value is 14.

Checks and common pitfalls: A radian argument is essential for the familiar sine derivative formula.

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

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • Apply power, product, quotient and chain rules with elementary derivatives.
    • Which condition is essential in derivative operations?
    • Which rule is needed before substituting a point into a composite function?

    Board plan

    • Defining relation: Apply power, product, quotient and chain rules with elementary derivatives.
      (uv)′=u′v+uv′;[f(g)]′=f′(g)g′(uv)\prime=u\prime v+uv\prime;\quad [f(g)]\prime=f\prime(g)g\prime
    • Conditions: Quotient denominators stay nonzero; logarithms have positive real arguments and trigonometric derivatives use radians.

    Anticipated thinking

    • The derivative of a product is not the product of the derivatives.

    Assessment checklist

    • 1 mark: choose the correct representation and conditions.
    • 1 mark: establish the intermediate relation.
    • 1 mark: complete a connected calculation or proof.
    • 1 mark: interpret and check the conclusion.

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