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.