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is
the limit of
as
goes
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.