Derivative of a Polynomial Function

Overview

A short animation that visually demonstrates how to compute the derivative of a simple polynomial, e.g., $f(x)=ax^2+bx+c$. The viewer sees the function, the limit definition, the algebraic simplification, and the final derivative, reinforcing the concept of differentiation as a rate of change.

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Phases

#	Phase Name	Duration	Description
1	Intro	~3s	Title fades in, then the polynomial $f(x)=ax^2+bx+c$ appears centered.
2	Limit Definition	~6s	The limit definition $f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$ slides in, with a visual "zoom" on the numerator as $f(x+h)$ expands.
3	Algebraic Expansion	~8s	Step‑by‑step expansion of $f(x+h)$ and subtraction of $f(x)$ is shown with animated brackets and term‑highlighting.
4	Simplification & Cancellation	~6s	Common terms cancel, leaving $\frac{2ahx+ah^2+bh}{h}$. The factor $h$ is highlighted and cancelled, revealing $2ax+b+ah$.
5	Taking the Limit	~4s	As $h\to0$, the term $ah$ fades out, leaving the final derivative $f'(x)=2ax+b$.
6	Outro	~3s	The original function and its derivative are displayed side‑by‑side with a brief caption "Derivative of a quadratic" and then fade out.

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Layout

┌─────────────────────────────────────────────┐
│                 TITLE AREA                  │
├───────────────────────┬─────────────────────┤
│        MAIN AREA      │   SUPPORT AREA      │
│   (function & steps) │ (notations, limits)│
├───────────────────────┴─────────────────────┤
│               BOTTOM AREA                 │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Title	"Derivative of a Polynomial" (large font)	Fades in at start of Intro, stays throughout
Main	Central display of the polynomial, expanded expressions, and simplified forms. Uses color‑coded terms (e.g., $a$ in blue, $b$ in green, $c$ in gray).	Primary visual focus; terms animate in sync with narration
Support	Small side panel showing the limit definition, the variable $h$ approaching zero, and occasional arrows linking steps.	Appears during Phase 2–5, updates dynamically
Bottom	Final caption "Derivative: $f'(x)=2ax+b$" and a brief note "Power rule: $\frac{d}{dx}x^n = nx^{n-1}$".	Fades in during Phase 5, remains to end

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Notes

• Keep total runtime under 30 seconds; the durations above sum to ~30 s.
• Use smooth fade‑in/out and transform animations to emphasize algebraic manipulation.
• Color‑code each coefficient ($a, b, c$) consistently across all phases.
• No textual narration is required; visual cues and brief on‑screen labels convey the process.
• The scene must be a single Manim Scene class.