Answer
$\dfrac{2}{5}$
Work Step by Step
Formula to calculate the directional derivative: $D_uf=\nabla f(x,y) \cdot
u$
or, $D_uf=\nabla f(x,y) \cdot \dfrac{v}{|v|}$
At $(2,8)$
$D_uf (2,8)=(\dfrac{y}{2\sqrt{xy}},\dfrac{x}{2\sqrt{xy}}) \cdot (\dfrac{3}{5},\dfrac{-4}{5})$
$D_uf (2,8)=(1,\dfrac{1}{4}) \cdot (\dfrac{3}{5},\dfrac{-4}{5})$
Hence, $D_uf (2,8)=\dfrac{2}{5}$
