Derivative Introduction

Overview

A brief visual introduction to the derivative as the instantaneous rate of change. The animation shows a parabola $f(x)=\frac{x^{2}}{2}$, a moving point, its tangent line, and how the slope evolves into the derivative formula \frac{d}{dx}\left(\frac{x^{2}}{2}\r\right)=x.

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Phases

#	Phase Name	Duration	Description
1	Title & Subtitle	~3 s	"What is a Derivative?" fades in, followed by the subtitle "Understanding Change Visually", then both fade out.
2	Axes & Curve	~4 s	Axes appear with numbered ticks, axis labels $x$ and $y$ are written, then the blue parabola $f(x)=\frac{x^{2}}{2}$ is drawn and the initial equation appears in the upper‑left corner.
3	Point, Tangent & Slope	~5 s	A yellow dot appears on the curve at $x=0.7$, the red tangent line is created, and a small slope label (e.g., "Slope=0.70") appears in the upper‑right corner. The text "Derivative = Instant Rate of Change" slides up from the bottom.
4	Motion Along Curve	~7 s	The dot travels smoothly from $x=0.7$ to $x=3.5$ over 6 s, the tangent line and slope label update continuously, illustrating how the slope changes.
5	Final Formula	~4 s	The bottom explanatory text fades out, the original equation transforms into the derivative formula \frac{d}{dx}\left(\frac{x^{2}}{2}\r\right)=x.
6	Outro / Pause	~3 s	A brief pause to let the final formula linger before the scene ends.

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Layout

┌───────────────────────────────────────┐
│               TOP AREA                 │
│   Title / Subtitle (centered)          │
├───────────────────────┬───────────────┤
│                       │               │
│        MAIN AREA      │   SIDE AREA   │
│   (Axes, curve, dot,  │   (optional   │
│    tangent, slope)   │   future use) │
│                       │               │
├───────────────────────┴───────────────┤
│               BOTTOM AREA              │
│   Equation / explanatory text (left)   │
└───────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Title "What is a Derivative?" and subtitle "Understanding Change Visually" (centered).	Fade‑in at Phase 1, fade‑out before Phase 2.
Main	Coordinate axes, the parabola $f(x)=\frac{x^{2}}{2}$, moving yellow dot, red tangent line, and dynamic slope label.	Primary visual focus throughout Phases 2‑5.
Side	Reserved for possible future annotations; left empty in this spec.	No content needed now.
Bottom	Upper‑left equation $f(x)=\frac{x^{2}}{2}$ that later transforms, and the bottom caption "Derivative = Instant Rate of Change".	Equation appears in Phase 2, caption appears in Phase 3, both fade/transform as described.

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Notes

• All transitions are smooth fades or creations; no abrupt jumps.
• The slope label updates in real time using the current $x$ value of the moving dot, formatted to two decimal places.
• The final transformation of the equation should keep the same screen corner (upper‑left) to emphasize continuity.
• Total runtime is approximately 26 seconds, well within the 30‑second guideline.
• Only one Scene class is required; the specification is designed for a single Manim Scene.