how would I partially differentiate 1-x/(x^2+y^2) with respect to y and -y/(x^2+y^2) with respect to x.
Hello,
$\displaystyle 1-\frac{x}{x^2+y^2}$
x is treated as a constant since you differentiate with respect to y.
So $\displaystyle \frac{\partial}{\partial y} \left(1-\frac{x}{x^2+y^2}\right)=-x \cdot \frac{\partial}{\partial y} \left(\frac{1}{x^2+y^2}\right)$
$\displaystyle =x \cdot \frac{2y}{(x^2+y^2)^2}$ by applying the chain rule (the derivative of $\displaystyle x^2+y^2$ with respect to y is $\displaystyle 2y$)
Same goes for the other one...