LEARN · EXPLAIN · REVISE

Find the derivative of x² at x=t from the limit.

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

高二選擇性必修第二册(A版).pdf · 5.1 · PDF 64 / printed page 59

TOPIC 01

Derivative: concept and meaning

Distinguish average and instantaneous rates and derive tangents from a limit.

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

01 / Foundation#Worked example

t=2

The difference quotient uses h≠0; differentiability requires a finite common two-sided limit.

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.

[(t+h)^2-t^2]/h=2t+h

Worked solution

Differentiate the specified function and substitute only after differentiating.
Apply the stated relation and retain its conditions.
$f'(2)=\lim_{h\to0}(4+h)=4$
Cancel h only while h≠0, then take the limit.

The requested value is 4.

Checks and common pitfalls: Cancel h only while h≠0, then take the limit.

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

Teacher preparation and assessment

Defining relation: Distinguish average and instantaneous rates and derive tangents from a limit.
$f\prime(a)=\lim_{h\to0}[f(a+h)-f(a)]/h$
Conditions: The difference quotient uses h≠0; differentiability requires a finite common two-sided limit.