Implicit functions, differentials, and their geometric interpretation

Consider the following equation:

Geometrically, this equation represents a sphere centered at $(0,0,0)$ with the radius $r=2$. The technical term for this is implicit surface because the relationship between $x$, $y$, and $z$ is defined implicitly.

The surface above can be also defined as:

This equation has the following geometric interpretation.

Consider a tangent plane touching the implicitly defined surface at some point $(x_0,y_0,z_0)$. The differential equation states that for any point $(x,y,z)$ on that plane, the following must hold:

or more concisely,

All such planes turn out to be tangent planes to spheres centered at $(0,0,0)$, and the $x=2,y=0,z=0$ initial condition allows us to select the "correct" sphere of radius $r=2$.