Answer
$\frac{\partial z}{\partial x}=f'(x)$
$\frac{\partial z}{\partial y}=g'(y)$
Work Step by Step
Given a function $z$ that is a sum of two single variable functions depending on $x$ and $y$ respectively such as: $z=f(x)+g(y)$ , the partial derivatives of $z$ can be found using the standard rules of derivatives:
First, differentiate w.r.t $x$. As such, all occurrences of the variable $y$ are considered constants.
$\frac{\partial z}{\partial x}=\frac{\partial}{\partial x} f(x) + \frac{\partial}{\partial x} g(y)=f'(x) + 0 = f'(x)$
The same can be done for the function w.r.t the variable $y$:
$\frac{\partial z}{\partial y}=\frac{\partial}{\partial y} f(x) + \frac{\partial}{\partial y} g(y)=0 + g'(y) = g'(y)$
