Partial Derivatives

Although this applies to higher dimensions too, this example sticks with a three variable equation because it can be visualized in three dimensional space.  It will look like a sheet that is tilted and/or bent in various ways.

Example equation

Graphed at geogebra

Partial derivative

This is essentially picking a point along one of the planes to get the equation for the line that is formed out of the sheet right there with respect to the other two planes.

Process

Starting with the original equation graphed above 

Pick a point along the Y axis.  In the graph above, the “sheet” crosses the Y axis at position 0, so using that it gives us this equation for the line that forms where the sheet passes that axis.

Deriving that like any other two variable equation gives this:

This is the slope along the z and x axes, it can be helpful to express it as 2/1.  If the image is turned to view edgewise along the sheet with respect to the z and x axes, this slope becomes observable: