Descartes Folium

The curve of equation (2) x³ + y³ = 6xy is called Descartes Folium.

Example 2 The diagram suggests that at the point (3, 3) the gradient of the tangent line is -1. We prove this by diﬀerentiating (2).

Solution. Differentiating x³ + y³ = 6xy with respect to x,

using the Chain Rule on the y³ term, and

the Product Rule on the 6xy term...

When x = 3 and y = 3, 2
y ' = 2-· 3---3 = - 1
32 - 2 · 3