




is
the limit of
asgoes
to 0.
The second derivative, ,
is the first y derivative of ;
this is the limit as first
then '
goes to zero, of .
Similarly
is another limit of exactly the same combination: the limit when '
goes to zero before .
When f is a "smooth" function of both x and y, this combination will
be arbitrarily close to both its limiting values when both and
'
are very small, and so the limiting values must be the same.