Partial Derivatives Are Two Slices

Here is the bowl, f of x one and x two equals x one squared plus x two squared, and a point a on the floor, at two and one half.

Hold x two fixed. The pink wall cuts the bowl along a curve, and the pink line touching that curve at a has slope four. That slope is the partial derivative of f with respect to x one.

The rest of the narration is in the book's Companion.