Implicit functions, differentials, and their geometric interpretation

Consider the following equation:

$x^2+y^2+z^2=4.$

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:

$xdx+ydy+zdz=0,x=2,y=0,z=0.$

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:

$x_0(x-x_0)+y_0(y-y_0)+z_0(z-z_0)=0,$

or more concisely,

$x_0\Delta x+y_0\Delta y+z_0\Delta z=0.$

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 $r=2$.