Recall that a circle is NOT a function.

Before differentiating, you could solve for $y$ and get

$y=\pm \sqrt{x^{2}-1}.$

Then find the derivative of each piece separately, or.....

You could use implicit differentiation!

Implicit differentiation:

$\frac{d}{dx}(x^{2}+y^{2})=\frac{d}{dx}(1)$

$\text{Now find }\frac{dy}{dx}.$