The complex functions $f(z)=u(x, y)+i v(x, y)$ and $\overline{f(z)}=u(x, y)-i v(x, y)$ are both analytic in a given domain. Choose the correct option(s) from the following.
$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}=0$
$\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x} \neq 0$
$\frac{d f(z)}{d z}=0$
$\frac{d f(z)}{d z} \neq 0$