$\displaystyle z=x sin^{-1}\left( \frac{x}{y} \right)$

determine $\displaystyle \left(\frac{\partial z}{\partial x}\right)_{y}$ and $\displaystyle \left(\frac{\partial z}{\partial y}\right)_{x}$

am I right in thinking that:

$\displaystyle x=ysin\left(\frac{z}{x}\right)$

and if so, where do I go from there? I don't suppose i can do this by diff w.r.t x:

$\displaystyle 1=\frac{y}{x}cos\left(\frac{z}{x}\right)\left(\fra c{\partial z}{\partial x}\right)_{y}$

giving me the rather dodgy looking answer:

$\displaystyle \frac{\partial z}{\partial x})_{y}= \frac{x}{ycos(z/x)}$

and for the other one, by diff w.r.t y:

$\displaystyle 0= sin(\frac{z}{x}) + ycos(\frac{z}{x}).\frac{\partial z}{\partial y})_{x}$

$\displaystyle \frac{\partial z}{\partial y})_{x}= -tan (\frac{z}{x})/y$

thanks, i hope at least the first step is right....