$f(x,y) =x^{2}y=12$
$ \begin{cases} \partial_{x}f = 2xy+x^{2}\dot{y} \\ \partial_{y}f = (2x \dot{x}) y + x^{2} \end{cases} $
now $\partial_{x}(2,3) =12+4\dot{y}$ and $\partial_{y}(2,3) =12\dot{x}+4$ but what are the terms $\dot{x}$ and $\dot{y}$? I need to calculate the change at point $x=2$ (so putting $x=2$ into $f$, I get point (2,3)). But I am unable make the leap here. Change in $f$ is? Is it a tuple $(\partial_{x}f, \partial_{y}f)$ or what does it mean? Help appreciated.