= Solution
First express the <function> in the original variables:
$$
f(x,y)=\ln(1+x^2+y^2)+\ln(1+x^2y^2).
$$
The <chain rule> gives
$$
f_x=\frac{2x}{1+x^2+y^2}+\frac{2xy^2}{1+x^2y^2}.
$$
Taking its <partial derivative> with respect to $y$ yields
$$
f_{xy}=-\frac{4xy}{(1+x^2+y^2)^2}+\frac{4xy(1+x^2y^2)-4x^3y^3}{(1+x^2y^2)^2}.
$$
The numerator in the second fraction reduces to $4xy$, reproducing
$$
\boxed{f_{xy}=4xy\left[\frac1{(1+x^2y^2)^2}-\frac1{(1+x^2+y^2)^2}\right].}
$$
Both <logarithms> are defined for every real $x,y$ because their arguments are at least one.
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