AnimG 标志AnimG

The Chain Rule

受众: University类别: Math

简介

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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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.

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

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

Assets & Dependencies

  • Fonts: LaTeX
  • Manim version: ManimCE 0.19.1

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