≥Better Grades
Menu
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

Problem

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

Complete worked solutions

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

01

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

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

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

  1. Write the composition as nested layers from outside to inside.
  2. Differentiate each layer once and multiply the factors.
  3. Therefore the result is y′=16(2x−3)(1+(2x−3)2)3y'=16(2x-3)(1+(2x-3)^2)^3.

What is included

See a concise answer and full derivation for differentiate y equals (1 plus (2 x minus 3) squared) to the power (4).

Skills assessed

  • Nested chain rule

Prerequisites

  • basic derivative rules
  • function composition

Common errors

  • Skipping a middle layer of the composition.

Learn fully

Supporting steps