References: Implicit differentiation

Source material

Source curriculum (structural mirror, cited as further study):
• 3Blue1Brown, Essence of Calculus, Chapter 7: "Implicit differentiation, what's going on here?"
  Creator: Grant Sanderson
  Lesson page: https://www.3blue1brown.com/lessons/implicit-differentiation
  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

• Implicit differentiation, what’s going on here? (3Blue1Brown) by Grant Sanderson. The video this lesson mirrors. Sanderson is especially careful here about why the procedure works rather than treating it as a recipe, framing it through tiny nudges to both x and y that must keep the relation satisfied. The sliding-ladder related-rates setup is also worked visually. About eleven minutes.

Going deeper

• Essence of Calculus (full series) by 3Blue1Brown. The series this track follows. The chain rule from earlier is what makes implicit differentiation work; the next chapter (Limits and the definition of derivatives) returns to the limit foundation underneath every derivative in the track so far.

• Khan Academy: Calculus for a slower, exercise-driven treatment of implicit differentiation and related rates, with practice problems and immediate feedback.

Adjacent topics

Where this sits in the track and the wider curriculum.

• The chain rule (earlier lesson). Implicit differentiation is the chain rule applied to a relationship: every y term is a composition (function of y(x)), so differentiating it deposits a dy/dx. The hyperbola example cross-checks against the power rule’s d/dx(1/x) = -1/x^2.

• Constrained optimization (Track 11 and beyond). Methods that optimize subject to constraints, and layers defined as fixed-point equations (deep equilibrium models), compute gradients through relationships that are not solved explicitly. Implicit differentiation is the calculus underneath those techniques; this lesson is where the idea is introduced cleanly on circles and ladders before it reappears in machine learning at scale.