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