Answer
$g_r=cosθ+sinθ$
$g_θ=-rsinθ+rcosθ$
Work Step by Step
Take the first partial derivatives of the given function. When taking partial derivative with respect to r, treat $θ$ as a constant, and vice versa:
$g_r=(1)cosθ+(1)sinθ=cosθ+sinθ$
$g_θ=-rsinθ+rcosθ$
