In 1712 the Royal Society named Isaac Newton the inventor of calculus, in a report Newton had secretly written himself. The prize was the derivative: a way to measure how fast anything changes at a single instant. This video builds it from scratch.
Chapter 3 of Thomas’ Calculus, told as one story: from the speed of a thrown ball at a single instant to the rules that differentiate almost any formula, and the three big payoffs: motion, related rates and linear approximation.
Every idea is met three ways: as something physical (a ball, a mountain road, a cafe’s revenue, a bicycle chain, a ladder), as an animated picture (secants swinging into tangents, a slope meter, growing squares and rectangles, number lines stretching), and as mathematics with full derivations.
In this video:
• The derivative at a point as a limit of difference quotients (tangent slope and instantaneous rate)
• The derivative as a function, notation (Newton, Leibniz, Lagrange) and where derivatives fail
• Why differentiability implies continuity, and why the converse is false
• Power, constant, sum, product and quotient rules, with geometric pictures and proofs
• Higher derivatives, velocity, acceleration, jerk and marginal cost
• The derivatives of sine and cosine from the unit circle and from limits (and why radians)
• The chain rule: a bicycle drivetrain, stretching number lines, and a careful proof
• Implicit differentiation and related rates (the sliding ladder)
• Linearization, differentials, and how fast the approximation error shrinks
How it teaches: Concrete → Pictorial → Abstract. Each idea starts with something physical you can picture, becomes a diagram you can see, and only then becomes notation and proof.
Chapters
0:00 The war over the derivative
0:28 Introduction
0:59 3.1 The derivative at a point
2:51 3.2 The derivative as a function
5:00 3.3 Power, constant and sum rules
6:46 3.3 Product and quotient rules
8:38 3.4 Velocity and acceleration
10:06 3.5 Derivatives of sine and cosine
11:49 3.6 The chain rule
13:44 3.7 Implicit differentiation
14:59 3.8 Related rates
16:44 3.9 Linearization and differentials
18:50 Recap and a puzzle
An independent companion to Chapter 3 (Derivatives) of Thomas’ Calculus, 14th edition (Hass, Heil & Weir). Not affiliated with or endorsed by the publisher.
Sources and further reading:
• G. W. Leibniz, Nova methodus pro maximis et minimis, Acta Eruditorum (1684)
• I. Newton, The Method of Fluxions and Infinite Series (written 1671, published 1736)
• J.-L. Lagrange, Théorie des fonctions analytiques (1797)
• A. Rupert Hall, Philosophers at War: The Quarrel between Newton and Leibniz (1980)
• R. S. Westfall, Never at Rest: A Biography of Isaac Newton (1980)
• Hass, Heil & Weir, Thomas’ Calculus, 14th edition, Chapter 3 (scope and order of topics)
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