LEARN · EXPLAIN · REVISE

Find f′(0).

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

TOPIC 01

Derivative operations

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

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

01 / Standard#Worked example

f(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\prime(x)=te^{tx}

Worked solution

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.

Teacher preparation and assessment

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