How to find $\frac{dy}{dx}$ for $\sqrt{xy} = 1$?

Question

What approach would be ideal in finding $\frac{dy}{dx}$ for $\sqrt{xy} = 1$?

Answer 1

Either solve for $y$ and differentiate, or you can use implicit differentiation:

$$ \sqrt{xy} = 1 \iff \frac{1}{2 \sqrt{x}}\sqrt{y} + \frac{1}{2 \sqrt{y}} \sqrt{x} \cdot y' = 0$$

by Product rule and chain rule, now you should be able to solve for $y'$

Answer 2

The square root can be ignored, just use $xy=1$. Then you can either differentiate implicitly $$y+xy'=0\Rightarrow y'=-\frac{y}{x}=-\frac{1}{x^2}=-y^2$$ or express explicitly $$y=\frac{1}{x}$$ and take the derivative of that.