Worked problem · Calculus I · Unit 2A

Nested Chain Rule: Worked Example and Solution

A complete nested chain rule example with method choice, derivation, verification, and a common wrong approach.

1 problem8 min estimated timeFoundational to advanced progression

What is included

See a concise answer and full derivation for differentiate $y=(1+(2x-3)^2)^4$.

Skills assessed

Nested chain rule

Prerequisites

basic derivative rules
function composition

Printable preview

Differentiate $y=(1+(2x-3)^2)^4$ .

Complete worked solutions

Every problem has a source-matched answer and independently reviewed derivation.

Problem 1: Differentiate $y=(1+(2x-3)^2)^4$ .

Answer: $y'=16(2x-3)(1+(2x-3)^2)^3$

Why this method: Nested chain rule matches the mathematical structure before any algebraic cleanup.

Write the composition as nested layers from outside to inside.
Differentiate each layer once and multiply the factors.

Common errors

Skipping a middle layer of the composition.

Complete worked solutions

Every problem has a source-matched answer and independently reviewed derivation.

Problem 1: Differentiate $y=(1+(2x-3)^2)^4$ .

Answer: $y'=16(2x-3)(1+(2x-3)^2)^3$

Why this method: Nested chain rule matches the mathematical structure before any algebraic cleanup.

Write the composition as nested layers from outside to inside.
Differentiate each layer once and multiply the factors.

What is included

Skills assessed

Prerequisites

Printable preview

Complete worked solutions

Problem 1: Differentiate y=(1+(2x−3)2)4y=(1+(2x-3)^2)^4y=(1+(2x−3)2)4.

Common errors

Continue learning

Lessons and articles

Related resources

Glossary

What is included

Skills assessed

Prerequisites

Printable preview

Complete worked solutions

Problem 1: Differentiate y=(1+(2x−3)2)4y=(1+(2x-3)^2)^4y=(1+(2x−3)2)4.

Common errors

Continue learning

Lessons and articles

Related resources

Glossary

Problem 1: Differentiate $y=(1+(2x-3)^2)^4$ .

Problem 1: Differentiate $y=(1+(2x-3)^2)^4$ .