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?
Since $\omega$ is defined as $f$, then $\partial f/\partial x = \partial \omega/\partial x$. In general: $$ \frac{\partial \omega}{\partial x} = \frac{\partial f}{\partial x},\quad \frac{\partial \omega}{\partial y} = \frac{\partial f}{\partial y} $$ $$ \frac{\partial \omega}{\partial t} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial t},\quad \frac{\partial \omega}{\partial r} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial r}. $$