What's the $\left \| \text{Jacobian matrix} \right \|$ of $f(x,y)=\frac{x}{y}$?

Question

Given is $f(x,y)=\frac{x}{y}$

Determine $\left\| J \right\|_{X,Y}$

I'm not sure if I did it correctly?

So we first derivate to $x$: $f(x,y)' = \frac{1}{y}$

Now derivate to $y$: $f(x,y)' = -\frac{x}{y^2}$

And now we somehow build the Jacobian matrix with these.. but how?

It should be $J= \begin{pmatrix} \frac{1}{y} & -\frac{x}{y^2} \end{pmatrix}$

If this is correct, then $\left\| J \right\|$ should be $\sqrt{({\frac{1}{y})^2}+(-\frac{x}{y^2})^2}$ ?