The functions $A(x,y)$ and $B(x,y)$ are defined by the following equation system:
$$x+y^2+2AB = 0 \text{ and } x^2-xy+y^3+A^2+B^2 = 0$$
Calculate the partial derivatives of $A_x, A_y, B_x$ and $B_y$.
I do not understand what I am supposed to do with this as I know nothing about the functions $A$ and $B$. Could you help me?
To calculate partial derivatives of $A$ and $B$, you only need to know that they are functions of the variables $x$ and $y$, as assumed in the question with $A(x,y)$ and $B(x,y)$.
Now take the partial derivative of the two given equations implicitly with respect to $x$,
$$1+2BA_x + 2AB_x = 0$$ $$2x-y+2AA_x+2BB_x=0$$
which is a system of linear equations in $A_x$ and $B_x$ and can be solved for the solutions below,
$$A_x=\frac{B-(2x-y)A}{2(A^2-B^2)},\>\>\>\>\>B_x=\frac{(2x-y)B-A}{2(A^2-B^2)}$$
Likewise, you can calculate $A_y$ and $B_y$ as well.