There Is No y = f(x) Here, and the Slope Still Exists

You'll learnto differentiate a relation exactly as it is written, to see why every y-term hands back a dy/dx factor for the same reason the chain rule always does, and to read a slope formula that needs both coordinates.

Draw a circle and run a vertical line through it. The line hits twice, so there is no function y = f(x) to differentiate — and yet the curve has a perfectly good tangent at both of those points. Implicit differentiation is how you get them. You differentiate the equation as it stands, and because y depends on x, every y-term is a composition: y² hands back 2y·dy/dx for exactly the reason u² hands back 2u·du/dx. Leave that factor off and you have not made a small error; you have differentiated a different equation.

8 STEPS · 6 QUICK CHECKS · x-TERMS ARE PLAIN, y-TERMS CARRY A FACTOR · ONE FACTOR PER LAYER, STILL · v1

d/dx[ y ⁿ ] = n y ⁿ⁻¹ · dy/dx · every y-term is a composition

Before you start

What you're looking at

The circle x² + y² = 25, and a vertical line about to sweep across it. Watch where the line meets the curve twice — that is a relation, not a function, and there is no y = f(x) to differentiate.

The question

If you cannot solve for y, how do you get the slope? And why does every y-term come back with a dy/dx attached to it?

Watch for

One factor per layer, exactly as in 3.1 — and step 7, where leaving that factor off one term draws a line that plunges through the curve instead of touching it.

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