The Derivative: How Calculus Finds the Slope of a Curve at a Single Point скачать в хорошем качестве

The Derivative: How Calculus Finds the Slope of a Curve at a Single Point

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 ───────────────────────────── ORIGINAL SOURCE ───────────────────────────── This video distills concepts from:    • Calculus 1 Lecture 2.1:  Introduction to t...   Full credit to the original creator for the educational content. ───────────────────────────── About Lecture Distilled ───────────────────────────── Long lectures. Short videos. Core insights. We distill lengthy academic lectures into focused concept videos that capture the essential ideas. Perfect for review, preview, or filling gaps in your understanding. Explore more: https://github.com/Augustinus12835/au... #calculus #derivatives #mathematics #limits #tangentline #calculusbasics #mathexplained #differentialcalculus