Below, we consider a point A(a,b) in the xy-plane and a point P(a,b,f(a,b)) on the surface. If you slide a and b, you can see how the tangent plane changes. If you activate u, e, and i, and then slide j and k, try to understand how this relates to directi
Below if f(x,y) = xy. Note that f_{xy} = f_{yx} = 1. The second partial f_{xy} measures the change in x-slopes as you move in the y-direction, and f_{yx} measures the change in y-slopes as you move in the x-direction.
Below is f(x,y) = ax^2+bxy+cy^2. Note that f_{xx}=2a, f_{yy}=2c, f_{xy} = b, so D = 4ac-b^2. Slide a, b, c around to see the change in the surface as these partials change. D is at the bottom if you want to see the number itself.