The Chain Rule

Description

Visualizes the chain rule through three stacked number lines representing the input x, the intermediate value g(x), and the output f(g(x)). Animated small changes Δx propagate through the composition, making the multiplicative structure of the chain rule visually intuitive.

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Phases

#	Phase Name	Duration	Description
1	Introduction	5s	Title, show three labeled number lines stacked vertically
2	Function Composition	6s	Animate a point x mapping to g(x) then to f(g(x)) with arrows
3	Small Change Δx	7s	Introduce Δx perturbation on bottom line, show how it propagates
4	Magnification Effect	8s	Show Δg ≈ g'(x)·Δx and Δ(f∘g) ≈ f'(g(x))·Δg visually
5	Chain Rule Formula	8s	Derive and display (f∘g)' = f'(g(x))·g'(x)
6	Concrete Example	8s	Work through d/dx sin(x²): g=x², f=sin, g'=2x, f'=cos
7	Summary	5s	Display chain rule formula prominently with the example result

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Layout

+--------------------------------------------------+
|  Title: "The Chain Rule"                         |
+--------------------------------------------------+
|                                                  |
|  ←────────────── Number line x ──────────────→  |
|                        ↕ g                       |
|  ←────────────── Number line g(x) ───────────→  |
|                        ↕ f                       |
|  ←───────────── Number line f(g(x)) ─────────→  |
|                                                  |
|  Right panel: chain rule formula                 |
|  d/dx f(g(x)) = f'(g(x))·g'(x)                 |
|                                                  |
+--------------------------------------------------+

Area Descriptions

• Main area: Three stacked horizontal number lines with mapping arrows
• Left labels: "x", "g(x)", "f(g(x))" identifying each line
• Right panel: Formula panel building up the chain rule equation
• Highlighted deltas: Δx, Δg, Δ(f∘g) shown as bracket intervals on lines

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Assets & Dependencies

• Fonts: LaTeX
• Manim version: ManimCE 0.19.1

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Notes

• Example: h(x) = sin(x²); g(x) = x², f(u) = sin(u)
• h'(x) = cos(x²) · 2x demonstrated at x=1 numerically
• Number lines use NumberLine objects with custom tick marks
• Δ brackets shown as braces (BraceBetweenPoints or Brace) with labels