References: The chain 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. The
previous Clawdemy lesson took the product-rule portion; this 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. The chain-rule portion shows the composition as nested nudges, where a change in x propagates through the inner function and then the outer one, the rates multiplying along the way. Seeing the “evaluated at the inner function” step in motion is the clearest way to avoid the classic error. About thirteen minutes (shared with the product rule).

Going deeper

• Essence of Calculus (full series) by 3Blue1Brown. The series this track follows. This lesson and the previous one split the product/chain-rule chapter; the next chapter (What’s so special about Euler’s number e?) examines the function that is its own derivative.

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

Adjacent topics

Where this sits in the track and the wider curriculum.

• The product rule (previous lesson). Product rule is for functions multiplied (f · g); the chain rule is for functions nested (f(g(x))). The two halves of one source chapter, and the two ways functions most commonly combine.

• Backpropagation (Track 11, neural network intuition). Track 11’s backpropagation lesson is this exact rule, applied through the layers of a neural network. A network is a deep composition of functions, and training it means differentiating that composition, which is the chain rule run backward layer by layer. The “rates multiply through a composition” framing here is precisely what “gradients flow backward through the layers” means there. These two lessons are reciprocal: this one is the math, that one is the application.