Define f(x,y) to be 
. Then for any unit vector 
, the directional derivative at the
origin is by definition
In particular, 
and 
(since these are directional
derivatives for 
and 
respectively). If f were
differentiable at the origin, then 
would equal 
for every 
. But
is not always zero, so f is not
differentiable.
Here's the graph of f:
