Answer
$\dfrac{4-3\sqrt 3}{10}$
Work Step by Step
Formula to calculate the directional derivative: $D_uf=f_x(x,y)a+f_y(x,y)b$
$D_uf=(e^x \sin y) \times \cos (\pi/3)+(e^x \sin y) \times \sin (\pi/3)$
This implies
At $(0,\dfrac{\pi}{3})$
$D_uf (0,\dfrac{\pi}{3})=\dfrac{-6}{10} \times \dfrac{\sqrt 3}{2}+\dfrac{8}{10} \times \dfrac{1}{2}$
Hence, $D_uf (0,\dfrac{\pi}{3})=\dfrac{4-3\sqrt 3}{10}$
