calculus - If $f(x,y)=sin^{-1}left(frac x yright)$ determine $left(frac{partial f}{partial x}right)_y$

I have attempted the question but I am not sure whether I am right. Here's what I have done:

$$sin(f)=\frac x y$$
Differentiating implicitly with respect to $x$, holding $y$ constant:
$$(\frac{\partial f}{\partial x})_ycos(f)=\frac1y$$
$$(\frac{\partial f}{\partial x})_y=\frac1{ycos(f)}$$
Using the trig identity:
$$(\frac{\partial f}{\partial x})_y=\frac1{y\sqrt{1-sin^2(f)}}$$

From the first equation for $sin(f)$:
$$(\frac{\partial f}{\partial x})_y=\frac1{y\sqrt{1-\frac {x^2}{y^2}}}$$

Is this correct or have I done something wrong?
My most fundamental question is: can I simply treat $f$ as another letter/variable like i have done in my attempt? Because it was written is $f(x,y)$ in the question. Is this just the same as $y(x)$ and $y$ for example?

Answer

I prefer to use the notation
$
\frac{\partial f}{\partial x}
$

and to recall that the derivative of $\sin^{-1} t$ (which is the inverse function of $\sin$, also denoted by $\arcsin$) is $(1-t^2)^{-1/2}$. We get then with the chain rule
$$
\frac{\partial f}{\partial x}=\bigl(1-x^2y^{-2}\bigr)^{-1/2} y^{-1}.
$$