The picture on the left shows a graph of

f(x,y) = |x|(1+y)

The red line shows the cross section x=0, while the green curve highlights the cross section y=0. Click and drag on the picture to rotate it; type "F" after clicking on the picture to view the cross sections without the surrounding surface.1

For this function,

  • fy(0,0)=0; it's the slope of the line tangent to the red curve at the point of intersection.
  • fx(0,0) does not exist, essentially for the same reason that y=|x| is not differentiable in single variable calculus. (In fact, it's exactly the same reason -- when y=0, f(x,y)=f(x,0)=|x|.)

Because fx(0,0) does not exist, f(x,y) is not differentiable at (0,0).

A function can fail to be differentiable at a point even if its partials exist at that point.

1 Why F? The applet on this page uses F to toggle the display of polygon Faces, which are the computer graphics objects which make up the displayed surface.