Answer
$$\dfrac{4-3\sqrt 3}{10}$$
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: $f(x,y)=e^x \sin y$
$D_uf(x,y)=(e^x \sin y) \times \cos (\pi/3)+(e^x \sin y) \times \sin (\pi/3)$
From the given data, we have : $(x,y)=$ $(0,\dfrac{\pi}{3})$
$D_uf (0,\dfrac{\pi}{3})=(\dfrac{-6}{10})(\dfrac{\sqrt 3}{2})+(\dfrac{8}{10})(\dfrac{1}{2})=\dfrac{4-3\sqrt 3}{10}$
