If you mean the total derivative, then it's just the vector of partial derivatives (called the gradient) of f, in this case (2x, 2y).

The total derivative is written as a vector, but you should think of it as a linear map from R² to R. Conceptually the total derivative is supposed to be the best linear approximation to f at a point, and since f is a function from R² to R, the total derivative must also be a function from R² to R. Linear functions from R² to R can be written as vectors. If (a,b) is a vector, and (x,y) is any element of R², the by taking the dot product of (a,b) and (x,y) we get a linear map from R² to R.

So for your function z = x² + y², the gradient is (2x,2y), and if we evaluate the gradient at a point (a,b) we get (2a,2b). Therefore the linear function g(x,y) which best approximates f(x,y) at (a,b) would be g(x,y) = (2a,2b).(x,y) = 2ax + 2by.