The surface f of x and y equals x squared y plus y, shown as a shaded map with contour lines, x from a quarter to three and three quarters and y from three quarters to five and a quarter; darker means higher. A draggable point sits on the map with a horizontal teal line through it, y frozen, and a vertical amber line, x frozen. To the right, two small graphs draw those slices: on top, f against x with y frozen, a parabola whose tangent slope at the point is two x y; below, f against y with x frozen, a straight line of slope x squared plus one. Readouts give f and both partial derivatives, and a nudge check compares the actual change in f for a step of one hundredth in x with the partial times the step.

One knob at a time

The surface f(x, y) = x²y + y as a map — darker is higher. Drag the point; the two slices through it are ordinary curves, and their slopes are the partials.