Answer
$$\dfrac{2}{5}$$
Work Step by Step
Our aim is to determine the directional derivative. In order to find it we have to use the expression:
$D_uf(x,y)=f_x(x,y)m+f_y(x,y)n$
or, $D_uf(x,y)=\nabla f(x,y) \cdot u=\nabla f(x,y) \cdot \dfrac{v}{|v|}$
Given: $f(x,y)=\sqrt{xy}$
From the given data, we have $(x,y)=$ $(2,8)$
$D_uf (2,8)=(\dfrac{y}{2\sqrt{xy}},\dfrac{x}{2\sqrt{xy}}) \cdot (\dfrac{3}{5},\dfrac{-4}{5})$
This yields, $D_uf (2,8)=(1,\dfrac{1}{4}) \cdot (\dfrac{3}{5},\dfrac{-4}{5})=\dfrac{2}{5}$
