Answer
$\dfrac{23}{42}$
Work Step by Step
$\nabla h=\biggl<\dfrac{3}{3r+6s+9t}, \dfrac{6}{3r+6s+9t}, \dfrac{9}{3r+6s+9t}\biggr>$
$\nabla h(1, 1, 1)=\biggr<\dfrac{1}{6}, \dfrac{1}{3}, \dfrac{1}{2}\biggr>$
$\hat{v}=\biggl<\dfrac{2}{7}, \dfrac{6}{7},\dfrac{3}{7}\biggr>$
$\mathbf{D_u}h(1, 1, 1)=\nabla h(1, 1, 1)\cdot\hat{v}=\dfrac{2}{42}+\dfrac{2}{7}+\dfrac{3}{14}=\dfrac{23}{42}$
