Find the gradient of $\frac{x}{x-y}$ It seems simple on the face of it, but I cannot figure out how to actually do this. I know that you have to find the partial with respect to $x$ and also with respect to $y$, but that's where I get lost.

$$\left.\begin{array}{rcl}\frac{\partial}{\partial x}\left( \frac{x}{x-y}\right) = \frac{(x-y)-x}{(x-y)^2} &=&\frac{-y}{(x-y)^2}\\
\frac{\partial}{\partial y} \left(\frac{x}{x-y}\right) &=& \frac{x}{(x-y)^2}\end{array}\right\} \Longrightarrow \nabla \left(\frac{x}{x-y}\right) = \frac{1}{(x-y)^2}\begin{pmatrix}- y \\ x\end{pmatrix}$$
Both times, you should use the derivation rule:
$$\frac{d}{dz}\left(\frac{f(z)}{g(z)}\right) = \frac{f'(z)g(z)-g'(z)f(z)}{g^2(z)},$$
one time with $f(z) = z, g(z) = z-y$ and the other time with $f(z) = 1, g(z) = x-z$
