## Calculus (3rd Edition)

$$f_{yy}(2,3)=- \frac{4}{9}.$$
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, $$f_{yy}(2,3)=- \frac{4}{9}.$$