Play: Partial Derivative Explorer
A function of two variables is a landscape — a height for every (x, y). We draw it as a 3-D surface, exactly like NMA: the height is the value (and it's shaded by color too — blue = low, yellow = high).
Three surfaces below, for whichever function you pick:
1. The function itself.
2. ∂f/∂x — the slope if you walk in the x-direction.
3. ∂f/∂y — the slope if you walk in the y-direction.
Switch functions and watch how the two slope-surfaces change shape.
Each surface is a 3-D plot — height(and color) is the value at each (x, y), just like NMA's plot3d. The second surface is the slope along x (∂f/∂x); the third is the slope along y (∂f/∂y). Notice: with the cross term 2xy, both partial-derivative surfaces tilt along x and y. Drop the cross term (x² + y²) and each partial becomes a flat ramp that only tilts along its own variable.
The thing to notice: for x · y and x² + 2xy + y² — the ones where x and y are multiplied together — the slope-map for x still has y in it (its color changes as you move up/down too). For x² + y² — no mixing — the x-slope map only changes left-to-right, and the y-slope map only changes top-to-bottom. That's the cross term at work, exactly as predicted.
Connect it back: remember the neuron's 1-D gain bump from the transfer-function lesson? A 2-D neuron (excitation + inhibition) has a gain surface for each input — these partial-derivative maps are exactly that, one level up.