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The Chain Rule

Overview

Functions inside functions.

What is it?

A formula for computing the derivative of the composition of two or more functions.

History

The key that unlocks complex physics, like modeling how air pressure changes as you climb a mountain (pressure depends on height, height depends on time).

Key Idea

Derivative of the OUTSIDE (keep inside same) * Derivative of the INSIDE.

Practice This Topic

Concept Guide

Plain English: Like peeling an onion. To find the rate of change of a complex function, you start with the outer layer and work your way in. It multiplies the rates together.

Real-world example: Blowing up a balloon. The volume depends on the radius, but the radius depends on how long you've been blowing. Chain rule links Time -> Radius -> Volume.

How to do it

  1. Identify the 'Outer' function (e.g., something squared) and 'Inner' function (e.g., 2x+1).
  2. Take derivative of Outer, leaving Inner alone.
  3. Multiply that result by the derivative of the Inner.
  4. Simplify.
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Common Pitfall

Stopping too early. Students often differentiate the outside function but forget to multiply by the derivative of the stuff inside.

Word Problem
"Air is being pumped into a spherical balloon. The radius is growing at 2 cm/sec. How fast is the Volume changing when the radius is 10 cm? (V = 4/3 π r³)."
Reasoning: We need dV/dt. Chain rule: dV/dr * dr/dt. Derivative of V is 4πr². We know dr/dt is 2. So, 4π(10)² * 2 = 800π cm³/sec.

Practice Examples

Logic
d/dx f(g(x)) = f'(g(x)) * g'(x)
Outside' times Inside'.