This demo shows graphs of f(x, y, z), one first partial derivative, and one second partial derivative. Red indicates a multiple of 5 (0 in this example). A yellow region next to a red one has a value greater than that multiple of 5, while a magenta region has a lower value.
The demo starts out with the example fzx(x, y, z). To change the first derivative, change g(x, y, z) to f_x(x, y, z), f_y(x, y, z), or f_z(x, y, z). The second derivative, h(x, y, z), will be some derivative of the first derivative with respect to x, y, or z, chosen the same way as before. For example, to see fxy(x, y, z), change g(x, y, z) to f_y(x, y, z) and h(x, y, z) to g_x(x, y, z).
Try changing x0, y0, and z0 to see different layers of these graphs and also to see how their values change as x, y, and z change. (For example, if the second derivative shown is fxy, notice how this second derivative graph describes the changes in fy as you change x0.)