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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?

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3nondatur
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    Assume $A$ and $B$ are defined on an open subset and satisfy the equations given. Then, for example, $1 + 2A_xB+2AB_x=0$ by taking the partial derivative with respect to the first factor. You do the rest. – William M. Oct 21 '19 at 21:19

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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.

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