Calculus I · 2A · quiz

Chain Rule Concept Quiz

Test conceptual understanding of chain rule concept quiz with a short self-check and model responses.

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Section overview

The chain rule and compositions

What this section is building

Test conceptual understanding of chain rule concept quiz with a short self-check and model responses.

Notice

A small input change passes through a sequence of machines; the total response multiplies the response at each stage.

Decide

List the layers, differentiate one layer at a time, and stop only when every input-dependent layer contributes.

Avoid

Differentiating the outside and leaving the inside unchanged without its derivative factor.

Concept Quiz: Chain Rule

• Why do derivative rates multiply through an intermediate variable? • Identify all layers in $\sqrt{1+\sin^2(3x)}$ . • What is missing from the incorrect derivative $(x^2+1)^5\mapsto5(x^2+1)^4$ ? • Which rule is applied first to $x^2\cos(x^3)$ ? • Why is $(\ln u)'=u'/u$ ? • Give two equivalent forms for the derivative of $e^{x^2}/x$ .

Structured concept quiz

Chain Rule Concept Quiz

Write a response before revealing the model. These conceptual items use an honest attempt-and-reveal rubric rather than pretending an open response has one machine-provable wording.

Interactive checkchain-rule-concept-quiz-01

Why do derivative rates multiply through an intermediate variable?

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkchain-rule-concept-quiz-02

Identify all layers in $\sqrt{1+\sin^2(3x)}$ .

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkchain-rule-concept-quiz-03

What is missing from the incorrect derivative $(x^2+1)^5\mapsto5(x^2+1)^4$ ?

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkchain-rule-concept-quiz-04

Which rule is applied first to $x^2\cos(x^3)$ ?

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkchain-rule-concept-quiz-05

Why is $(\ln u)'=u'/u$ ?

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkchain-rule-concept-quiz-06

Give two equivalent forms for the derivative of $e^{x^2}/x$ .

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Concept Quiz: Chain Rule

• Why do derivative rates multiply through an intermediate variable? • Identify all layers in

\sqrt{1+\sin^2(3x)}

. • What is missing from the incorrect derivative

(x^2+1)^5\mapsto5(x^2+1)^4

? • Which rule is applied first to

x^2\cos(x^3)

? • Why is

(\ln u)'=u'/u

? • Give two equivalent forms for the derivative of

e^{x^2}/x

Structured concept quiz

Chain Rule Concept Quiz

Write a response before revealing the model. These conceptual items use an honest attempt-and-reveal rubric rather than pretending an open response has one machine-provable wording.

Interactive checkchain-rule-concept-quiz-01

Why do derivative rates multiply through an intermediate variable?

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkchain-rule-concept-quiz-02

Identify all layers in $\sqrt{1+\sin^2(3x)}$ .

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkchain-rule-concept-quiz-03

What is missing from the incorrect derivative $(x^2+1)^5\mapsto5(x^2+1)^4$ ?

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkchain-rule-concept-quiz-04

Which rule is applied first to $x^2\cos(x^3)$ ?

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkchain-rule-concept-quiz-05

Why is $(\ln u)'=u'/u$ ?

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Interactive checkchain-rule-concept-quiz-06

Give two equivalent forms for the derivative of $e^{x^2}/x$ .

Your work stays on this device. No account or AI grader is used.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Source & rights

Original instruction with traceable references.

BetterGrades-original composition declared by source handoff; owner provenance review required before public release

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.