To start I find the derivative with respect to fx = 2y + x
derivative with respect to fy = 2 x - e^x/((e^x + 2)^2 (1/(e^x + 2)^2 + 1)) + y
Then to find the answer multiply fx by fy.
However the correct answer is 1 and apparently it can be found without writing anything down. How is that possible?
Take advantage that the arctan function depends only on x
when you differentiate according to y it equals zero
So we have that:
$$g(x,y)=\arctan(\frac 1 {e^x+2})+x^2+xy+y^2$$ differentiate wrt x
$$\partial_x g=\partial_x (\arctan(\frac 1 {e^x+2})) +2x+y$$ differentiate wrt y $$\partial_{yx} g=0+0+1=1$$