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The Chain Rule

The Hook

Functions inside functions.

What is it?

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

Time Travel (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).

Cheat Code

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

Practice This Now!

The Field Guide

In 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.

In The Real World: 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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Booby Trap!

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

Real World Challenge
"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)."
The Logic: 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.

Training Drills

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