Taylor's theorem for multivariable functions

[Outrageous]

For a function z=f(x,y),
Keeping y constant, using Taylor's
f(x+Δx,y)-f(x,y)=...-equation 1
f(x,y+Δy)-f(x,y)=...-equation 2
Then total change of z =equation 1 + equation 2
Is this correct?
but in the picture uploaded http://www.math.auckland.ac.nz/mathwiki/images/d/dc/MATHS_362_Semester_2_2007_Lecture_1.pdf
How to get that 2fxy(x-a)(y-b)?

Please guide.
Thank you

 

[mathman]

To get the total change you need terms that represent x and y changing together.
That leads to the term you asked about: ∂²f/∂x∂y

 

[Outrageous]

To get the total change you need terms that represent x and y changing together.
That leads to the term you asked about: ∂²f/∂x∂y

Thank you