Calculus I · Unit 2A · quiz

Chain Rule Concept Quiz

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 $e^{x^2}/x$ .

After the explanation

Use the section idea

Reading lens

Read nested functions from the outside inward, but multiply local response factors through every layer.

Mental model

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

Decision

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

Common trap

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

Check yourself

Can you annotate every factor in your derivative with the layer that produced it?

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?