PDE, change of variables and differential operator “transformation”

Given the wave equation:

$$\frac{\partial^2 f}{\partial t^2} = c^2 \frac{\partial^2 f}{\partial x^2}$$

I change variables in this way:

$$\xi = x+ct \\ \eta=x-ct$$

And the differential operators transform:

$$\frac{\partial }{\partial x} = \frac{\partial }{\partial \xi} + \frac{\partial }{\partial \eta} \\ \frac{\partial }{\partial t} = c^2 ( \frac{\partial }{\partial \xi} - \frac{\partial }{\partial \eta} )$$

I know this has something to do with the chain rule, but I'm not able to understand. How do I verify that these equalities are correct? How do I get to them?

Thank you.

Define $F(\xi,\eta)=f(t,x)$. Then, with some abuse of notation, $$\frac{\partial}{\partial x}f(t,x)=\frac{\partial}{\partial x}F(\xi,\eta)=\frac{\partial F}{\partial \xi}\frac{\partial\xi}{\partial x}+\frac{\partial F}{\partial \eta}\frac{\partial\eta}{\partial x}=\frac{\partial F}{\partial \xi}+\frac{\partial F}{\partial \eta}.$$ This is the meaning of $$\frac{\partial}{\partial x}=\frac{\partial }{\partial \xi}+\frac{\partial }{\partial \eta}.$$ A similar computation applies to the other derivative.