Implicit differentiation

What you’ll learn

Every derivative so far started from y = f(x), one variable written cleanly as a function of the other. But most relationships are not like that: the circle x² + y² = 25 ties x and y together without making either a clean function of the other, yet it has a definite slope at every point. The single capability this lesson builds: differentiate a relation that is not solved for y, and read the result as a slope (or, against time, as related rates).

The technique, implicit differentiation, introduces no new machinery: it is the chain rule applied to a relationship. Treat y as an unknown function of x, differentiate both sides, and attach a dy/dx to every y term (because each is a composition, d/dx(y²) = 2y·dy/dx), then solve algebraically. You will work the circle (dy/dx = -x/y, checked as perpendicular to the radius), handle a relation that genuinely cannot be untangled (x² + xy + y² = 7), derive the ln(x) derivative as 1/x from e^y = x, and meet related rates, the time-based twin, where differentiating a constraint with respect to time links two moving quantities’ rates (the sliding ladder).

Where this fits

This is lesson 8 of Phase 2 (The differentiation toolkit). It is built directly on the chain rule (lesson 6) and uses the self-derivative property of e (lesson 7) in the ln(x) derivation, so it ties the phase’s rules together. It is also a foundation the neural-network tracks return to: constrained optimization and fixed-point (deep equilibrium) layers are implicit differentiation at scale. Phase 2 closes with limits (lesson 9), and Phase 3 then turns to integration.

Before you start

Prerequisite (within this track): lesson 7, Why e is special (the ln(x) derivation uses e’s self-derivative property). The deeper tool this lesson rests on is the chain rule (lesson 6), since implicit differentiation is the chain rule applied to a relationship, so keep that fresh. You also want the product rule (lesson 5) for mixed terms like xy. Comfort solving a linear equation for one unknown (here, dy/dx) is the only algebra needed; no coding, nothing installed. The practice is pen and paper.

By the end, you’ll be able to

• Find dy/dx for a relation not solved for y by treating y as a function of x and differentiating both sides
• Explain why implicit differentiation is the chain rule (every y term is a composition that deposits a dy/dx)
• Apply the product rule to mixed terms like xy and solve the resulting equation for dy/dx
• Set up and solve a related-rates problem by differentiating a constraint with respect to time

Time and difficulty

• Read time: about 11 minutes
• Practice time: about 13 minutes (finding dy/dx for relations you cannot solve for y, a related-rates problem, and flashcards)
• Difficulty: standard