Answer
$-g(x)$ and $g(y)$
Work Step by Step
We need to take the first partial derivatives of the given function.
In order to find the partial derivative we will differentiate with respect to $x$, by keeping $y$ as a constant, and vice versa:
$f_x=\int_x^ y g(t) dt =- \int_{y}^{x} g(t) dt=\dfrac{\partial f}{\partial x}=-g(x)$
$f_y=\int_x^ y g(t) dt =\dfrac{\partial f}{\partial y}=g(y)$
