Answer
$$ f_{yy}(2,3)=- \frac{4}{9}.$$
Work Step by Step
Since $ f(x,y)=x\ln ( y^2)$, then by using the chain rule, we have
$$ f_y(x,y)=\frac{\partial f}{\partial y}=\frac{x}{y^2}(2y)=\frac{2x}{y}.$$
Again using the chain rule, we get
$$ f_{yy}(x,y)=\frac{\partial^2 f}{\partial y^2}=- \frac{2x}{y^2}.$$
Hence, $$ f_{yy}(2,3)=- \frac{4}{9}.$$
