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References: The derivative as a rate

Source material

Section titled “Source material”
Source curriculum (structural mirror, cited as further study):
• 3Blue1Brown, Essence of Calculus, Chapter 2: "The paradox of the derivative"
  Creator: Grant Sanderson
  Lesson page: https://www.3blue1brown.com/lessons/derivatives
  Series index: https://www.3blue1brown.com/?topic=calculus
  License: copyright Grant Sanderson; videos published on his site and YouTube
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Watch this next

Section titled “Watch this next”
  • The paradox of the derivative (3Blue1Brown) by Grant Sanderson. The video this lesson mirrors. Watching the secant line pivot into the tangent as the interval shrinks is the clearest way to see “rate at an instant” become “slope at a point,” and Sanderson’s car-distance example makes the average-to-instantaneous transition concrete. He is also careful, as this lesson is, that dt is a small ordinary number you let shrink, not a mystical infinitesimal. About eighteen minutes.

Going deeper

Section titled “Going deeper”

  • Essence of Calculus (full series) by 3Blue1Brown. The series this track follows. The previous chapter set up the rate-vs-accumulation duality; the next (Power Rule through geometry) finds the patterns that let you compute derivatives without expanding binomials each time.

  • Khan Academy: Calculus for a slower, exercise-driven treatment of limits, secants and tangents, and the definition of the derivative, with practice problems and immediate feedback.

Adjacent topics

Section titled “Adjacent topics”

Where this sits in the track.

  • The essence of calculus (previous lesson). That lesson glimpsed the rate side when the circle’s accumulated area πR² changed at the rate 2πR. This lesson makes that rate precise: what it means to measure how fast something changes at a single instant, via the limit of rise over run.

  • Power Rule through geometry (next lesson). This lesson computed the derivative of t^3 from scratch by expanding (t + dt)^3. That is tedious to repeat. The next lesson reveals the patterns, starting with the power rule, that let you read off the derivative of t^n and other common functions directly, with geometric reasoning rather than algebra.