The original utility function is \(\sqrt{xy}\). To find the slope or MRS, we have to calculate the first derivative, but because there are two variables, we have to find two partial derivatives—the derivative of just the \(x\) part while holding \(y\) constant and the derivative of just the \(y\) part while holding \(x\) constant—and then calculate the ratio of the two partial derivatives.
If you put derivative sqrt(xy) into
Wolfram Alpha, it will give you both partial derivatives:
\[ \begin{aligned} \frac{\partial}{\partial x} &= \frac{y}{2 \sqrt{xy}} & \text{[partial derivative of } x \text{ part]} \\ \frac{\partial}{\partial y} &= \frac{x}{2 \sqrt{xy}} & \text{[partial derivative of } y \text{ part]} \end{aligned} \]
To get the overall total derivative, we need to find the ratio of the partial \(x\) part to the partial \(y\) part, or \(\frac{\partial}{\partial x} / \frac{\partial}{\partial y}\). This looks like a huge mess, but it simplifies down a lot because of what happens when you divide by a fraction (you multiply by its inverse)—that whole \(2 \sqrt{xy}\) thing cancels out and disappears:
\[ \begin{aligned} \cfrac{\partial}{\partial x} / \cfrac{\partial}{\partial y} &= \frac{\ \cfrac{y}{2 \sqrt{xy}}\ }{\cfrac{x}{2 \sqrt{xy}}} \\[10pt] &= \frac{y}{2 \sqrt{xy}} \times \frac{2 \sqrt{xy}}{x} \\[10pt] &= \frac{y \times 2 \sqrt{xy}}{x \times 2 \sqrt{xy}} \\[10pt] &= \frac{y}{x} \end{aligned} \]