Answer
$\dfrac{-1-y^2}{(xy-1)^2}$ and $\dfrac{-1-x^2}{(xy-1)^2}$
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 $y$ as a constant.
$f_x=\dfrac{(xy-1)(1)-(x+y)(y)}{(xy-1)^2}=\dfrac{-1-y^2}{(xy-1)^2}$
Take the first partial derivatives of the given function $f(x,y)$ with respect to $y$, by keeping $x$ as a constant.
$f_y=\dfrac{(xy-1)(1)-(x+y) \times (x)}{(xy-1)^2}=\dfrac{-1-x^2}{(xy-1)^2}$
