Answer
$2+\dfrac{\sqrt 3}{2}$
Work Step by Step
Formula to calculate the directional derivative: $D_uf=f_x(x,y)a+f_y(x,y)b$
$D_uf=(-ye^{-x}) \times \cos (2\pi/3)+(e^{-x}) \times \sin (2\pi/3)$
This implies
At $(0,4)$
$D_uf(0,4)=-4 \times \dfrac{-1}{2}+1 \times \dfrac{\sqrt 3}{2}$
$D_uf(0,4) =2+\dfrac{\sqrt 3}{2}$
