Answer
$1$
Work Step by Step
$\nabla f=\biggl<2xy, x^2+2yz, y^2\biggr>$
$\hat{v}=\biggl<\dfrac{2}{3}, -\dfrac{1}{3}, \dfrac{2}{3}\biggr>$
Then,
$\nabla f_{(1, 2, 3)}=\biggl<4, 13, 4\biggr>$
Therefore,
$\mathbf{D_\hat{v}}f_{(1, 2, 3)}=\nabla f_{(1, 2, 3)}\cdot\hat{v}=\dfrac{8}{3} -\dfrac{13}{3}+\dfrac{8}{3}=1$
