Chain Rule Visualizer – AP Calculus AB | ShowMeClass

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Chain Rule Visualizer

AP Calculus AB · Unit 2: Derivatives

Learning Objectives

Understand the chain rule: d/dx[f(g(x))] = f'(g(x))·g'(x)

Visualize how "outer" and "inner" derivatives compose

Compare f(g(x)) with its tangent line at a point

Explore how changing x affects the chain of derivatives

Chain Rule Formula

Graph Legend

f(g(x)) — Composite function

g(x) — Inner function

f(u) — Outer function

Tangent line at x₀

Key Insight

The chain rule multiplies two rates of change: the rate at which g changes with x times the rate at which f changes with g. Think of it as a gear ratio — how fast the output spins depends on both gear stages.

Chain Rule Derivatives AP Calculus AB Unit 2

COMPOSITE f(g(x)) & TANGENT

INNER g(x) & OUTER f(u)

inner

g'(x₀)

×

outer

f'(g(x₀))

=

chain

(f∘g)'(x₀)

x₀ (evaluation point) 1.00

Outer Function f(u)

Inner Function g(x)

Chain Rule Computation

Display

📝 Quick Quiz▼

1. What is d/dx[sin(x²)]?

cos(x²) 2x·cos(x²) 2x·sin(x²) cos(2x)

2. If h(x) = e^(3x), what is h'(x)?

e^(3x) 3e^(3x) 3xe^(3x) e^(3x)/3

3. For f(g(x)), the chain rule says the derivative equals:

f'(x)·g'(x) f'(g(x))·g'(x) f(g'(x)) f'(g(x))+g'(x)