References: The product rule

Source material

Source curriculum (structural mirror, cited as further study):
• 3Blue1Brown, Essence of Calculus, Chapter 5: "Visualizing the chain rule and product rule"
  Creator: Grant Sanderson
  Lesson page: https://www.3blue1brown.com/lessons/chain-rule-and-product-rule
  Series index: https://www.3blue1brown.com/?topic=calculus
  License: copyright Grant Sanderson; videos published on his site and YouTube
Note: this 3B1B chapter covers both the product rule and the chain rule. This
Clawdemy lesson takes the product-rule portion; the next lesson takes the chain
rule from the same chapter.
Clawdemy's lessons are original prose that follows the pedagogical arc of this
series. We do not reproduce or transcribe the videos; we cite them as the
recommended companion. All rights to the original videos remain with the creator.

Watch this next

• Visualizing the chain rule and product rule (3Blue1Brown) by Grant Sanderson. The video this lesson mirrors. Watching the rectangle gain its two strips while the tiny corner shrinks away is the clearest way to see why the product rule has two terms and why f' · g' is the piece that does not survive. The same video also covers the chain rule, which the next lesson takes up. About thirteen minutes.

Going deeper

• Essence of Calculus (full series) by 3Blue1Brown. The series this track follows. The previous chapter found the trig derivatives used in this lesson’s examples; the next Clawdemy lesson takes the chain rule from this same chapter, and the chapter after that (What’s so special about Euler’s number e?) is where the series goes next.

• Khan Academy: Calculus for a slower, exercise-driven treatment of the product rule, with practice problems and immediate feedback.

Adjacent topics

Where this sits in the track.

• Trig derivatives (previous lesson) and the power rule (two lessons back). This lesson’s worked examples lean on both: x^2 · x^3 cross-checks against the power rule, and x · sin(x) and sin(x) · cos(x) use the trig derivatives. The rules are starting to cooperate.

• The chain rule (next lesson). The product rule handles functions that are multiplied; the chain rule handles functions that are nested, one inside another, like sin(x^2). Both come from the same 3B1B chapter and the same nudge-and-look reasoning; together they let you differentiate almost any combination of functions you will meet.