Derivatives: The Tangent Line

Description

Builds intuition for the derivative by animating the limit definition. A secant line between two points on f(x) = x² is drawn and animated as h approaches 0, visually converging to the tangent line. The difference quotient formula transforms into the derivative formula.

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Phases

#	Phase Name	Duration	Description
1	Setup	5s	Title, draw axes, plot f(x) = x² in blue
2	Secant Line	7s	Mark point (x, f(x)) at x=2, mark second point at x+h, draw secant
3	Difference Quotient	6s	Display rise/run formula, label Δx and Δy on graph
4	Animate h → 0	12s	ValueTracker h from 1.5 down to 0.01; secant morphs to tangent
5	Limit Formula	8s	Show full limit definition of derivative, highlight as h→0
6	Compute f'(2)	7s	Algebraically compute derivative of x², show f'(x) = 2x, f'(2) = 4
7	Summary	5s	Tangent line at x=2, slope label, geometric interpretation text

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Layout

+--------------------------------------------------+
|  Title: "Derivatives: The Tangent Line"          |
+--------------------------------------------------+
|                                                  |
|   Axes with f(x) = x²        | Formula panel    |
|   (left 60%)                 | (right 40%)      |
|                              |                  |
|   • Point P = (2, 4)         | f(x+h) - f(x)    |
|   • Point Q = (2+h, (2+h)²)  | ─────────────    |
|   • Secant line (orange)     |      h           |
|   → tangent line (yellow)    |                  |
|                              | → lim as h→0     |
|   Δx and Δy labels           | → f'(x) = 2x    |
|                              |                  |
+--------------------------------------------------+

Area Descriptions

• Left 60%: Axes with curve, points, secant/tangent lines, Δx and Δy visual indicators
• Right 40%: Evolving formula from difference quotient to derivative definition
• Below formula: Step-by-step algebraic simplification

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Assets & Dependencies

• Fonts: LaTeX
• Manim version: ManimCE 0.19.1

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Notes

• Use ValueTracker for h; always_redraw for secant line, Q point, and Δ labels
• x₀ fixed at x=2 for concreteness, f'(2)=4 is the expected tangent slope
• Color coding: curve = BLUE, secant = ORANGE, tangent = YELLOW, Δx = GREEN, Δy = RED