I'm having trouble understanding the chain rule for partial derivatives. If I'm given that $\omega=f(x,y)$ where $x$ and $y$ are functions of both $t$ and $r$, then by chain rule I can write that: $$\frac{\partial \omega}{\partial t}=\frac{\partial \omega}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial \omega}{\partial y}\frac{\partial y}{\partial t}$$
But if I'm asked to find out what $\frac{\partial f}{\partial x}$ is equal to then can I write that it's equal to $\frac{\partial \omega}{\partial x}?$ If I'm wrong then what is $\frac{\partial f}{\partial x}$ equal to?