Answer
$g_r=1-\cos \theta ; \\ g_θ=r \space \sin \theta ; \\g_z=-1$
Work Step by Step
Our aim is to take the first partial derivative of the given function $g(x,y)$ with respect to $r$, by keeping $z$ and $\theta$ as constants:
$g_r=1-\cos \theta$
Now, we repeat the process for the other partial derivatives:
$g_θ=r \space \sin \theta$
$\\g_z=-1$