The \(\dfrac{\mathrm dy}{\mathrm dx}\) row is filled by just plugging in the corresponding values of \(x\) and \(y\) into the ODE.
For example, when \(x=-1\) and \(y=1\), you have
\(\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{xy}3\implies\dfrac{\mathrm dy}{\mathrm dx}\bigg|_{x=-1,y=1}=\dfrac{(-1)(1)}3=-\dfrac13\)
and so on.
