Answer
$$y'=-\frac{y}{x}$$
Work Step by Step
$$x^2y^2=1$$
Using implicit differentiation, we differentiate both sides of the equation with respect to $x$: $$2xy^2+2x^2y\times y'=0$$
$$xy^2+x^2y\times y'=0$$
Now we set all elements with $y'$ to one side and the rest to the other side: $$x^2y\times y'=-xy^2$$
Finally, we calculate $y'$: $$y'=-\frac{xy^2}{x^2y}=-\frac{y}{x}$$
