Is Marginal Rate of Substitution a multivariable function? Suppose I have $U(x,y)$ and a level set of indifference curves. Suppose the value of $U$ along a given curve is $\bar{U}$. We know $dU = 0$. We compute total derivative, rearrange, and now have 

$$\frac{dy}{dx} = -\frac{U_x}{U_y}$$ 

My Confusion and Question:

$\frac{dy}{dx}$ is written as a function of a single variable right? But when we talk about $MU$, we always talk about it as a multivariable function. I thought $MRS$ was the ordinal utility analog of the cardinal utility $MU$. So why is $MRS$ a single variable function?
$\frac{dy}{dx}$ is written as a function of a single variable right?


I don't think so. Your shorthand notation should be standing for

$$\frac{dy}{dx} = -\frac{U_x(x, y)}{U_y(x,y} $$

and hence is a function of both variables ${x,y}$. That makes sense, as your MRS could depend on the locus you're at.