Скачать или смотреть Calculus 1 — 9.1: Derivative as Slope of a Curve

A single point has no rise and no run—so how can it have a slope? This video explains the fundamental insight behind derivatives: by sliding two points infinitely close together, a secant line becomes a tangent line, and the impossible becomes calculable.

You'll see the complete derivation from first principles, including a step-by-step calculation of the derivative of x² using the limit definition.

Key concepts covered:
• Why slope at a single point seems impossible and how limits solve it
• Secant lines vs. tangent lines and the transition between them
• The difference quotient: [f(x+h) - f(x)]/h
• The formal limit definition of the derivative
• Why we're not actually dividing by zero
• The derivative as a function, not just a number
• Complete worked example: finding f'(x) = 2x from f(x) = x²
• Algebraic checkpoints for verifying your derivative calculations

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SOURCE MATERIALS
The source materials for this video are from    • Calculus 1 Lecture 2.1:  Introduction to t...