LEARN · EXPLAIN · REVISE

Is |x−t| differentiable at x=t?

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 / Standard#Worked example

t=3

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

Compare the left and right difference quotients.

Hint 2

Use this intermediate relation.

|h|/h

Worked solution

The requested relation or conclusion is shown below.

f\prime(3)\text{ undefined}

Checks and common pitfalls: Continuity does not remove the mismatch in the one-sided derivatives.

Reasoning checklist · self / teacher assessment

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.