2 days ago I asked this question, I already got what I was asking there. Now I want to find the best approximations $P(a,b),Q(a,b)$ of $x(a,b),y(a,b)$ respectively, where $P,Q$ are two polynomials of degree $2$ in $a,b$.
What I have tried is to do so using Taylor series expansion for two variables. To do so I need the partial derivatives $\frac{{\partial x}}{{\partial a}}=1/22, \frac{{\partial x}}{{\partial b}}=1/11, \frac{{\partial y}}{{\partial a}}=1/11, \frac{{\partial y}}{{\partial b}}=-5/33$ that I have calculated derivating implicitly the functions $f_1:x^2y^3+x^3y^2+x^5y+1=a$ and $f_2:xy^2-2x^2y^4+3x^3y=b$.
Now I also need the partial derivatives $\frac{{\partial^2 x}}{{\partial a^2}}, \frac{{\partial^2 x}}{{\partial b^2}}, \frac{{\partial^2 y}}{{\partial a^2}}, \frac{{\partial^2 y}}{{\partial b^2}}$
To find them I suppose I have no other option but derivating implicitly again (is this true?), what seems very tedious.
Is this correct? If not, how do I do it?