$\sqrt{|xy|} = 1$

Attempting to find the derivative gives me $$\frac12(xy)^{-1/2}\left(x\frac{dy}{dx} + y\right) = 0$$

But I haven't figured out how to simplify this further. My teacher says that that's all I'll need to know, but I want to understand how the derivative of $\sqrt {xy} = 1$ is $-\frac{y}x$.

Edited to explain that I know the whole thing equals zero, but how do I solve for (dy/dx)?

Attempts to solve get me this far:
![enter image description here](https://i.stack.imgur.com/Sdv3t.png)