Answer
$y+z ; \\x+z ; \\y+x$
Work Step by Step
Our aim is to take the first partial derivatives of the given function $f(x,y)$ with respect to $x$, by keeping $y$ and $z$ as a constant, and vice versa:
$\dfrac{\partial }{\partial x}f(x,y)=y+z ; \\\dfrac{\partial }{\partial y}f(x,y)=x+z ; \\\dfrac{\partial }{\partial z}f(x,y)=y+x$