How do you take the derivative of arctan(y/x) with respect to x??
Why is this listed under "differential equations"?
Are we to assume that y is not a function of x? Then use the chain rule, of course. The derivative of arctan(u), with respect to u, is $\displaystyle \frac{1}{u^2+1}$. By the chain rule, with u= y/x, the derivative of arctan(y/x), with respect to x, is $\displaystyle \frac{1}{(y/x)^2+1}\frac{d(yx^{-1})}{dx}$. Of course, you differentiate $\displaystyle y/x= yx^{-1}$, with respect to x, by treating y as a constant.
To take the partial with respect to x, we treat y as a constant and since we have function composition we need the chain rule
$\displaystyle \frac{\partial }{\partial x}\tan^{-1}\left( \frac{y}{x}\right)=\frac{1}{1+\left( \frac{y}{x}\right)^2}\cdot \left( -\frac{y}{x^2}\right)=\frac{-y}{x^2+y^2}$