Let $ $ $ w (x,y) = f (y-x, x+y) $ , where $ f: \Bbb R^2 \to \Bbb R $ is a $ \mathcal C^2 $ classe function. Show that $$ 4\frac{\partial^2 f}{\partial u \partial v} = \frac{\partial^2 w}{\partial y^2} - \frac{\partial ^2 w}{\partial x^2 } $$ knowing that $ u=y-x $ and $ v=x+y . $
This is how far i could go:
$$ \frac{\partial w}{\partial x} = -\frac{\partial f}{\partial u} + \frac{\partial f}{\partial v} $$
$$\frac{\partial w}{\partial y}= \frac{\partial f}{\partial u} + \frac{\partial f}{\partial v} $$
If $$\frac{\partial w}{\partial x}=-\frac{\partial f}{\partial u}+\frac{\partial f}{\partial v},$$ then we find via the same process as you employed before that $$\frac{\partial^2 w}{\partial x^2}=\frac{\partial ^2f}{\partial u^2}-2\frac{\partial ^2f}{\partial u\partial v}+\frac{\partial^2 f}{\partial^2 v}.$$ Similarly, $$\frac{\partial ^2w}{\partial y^2}=\frac{\partial ^2f}{\partial u^2}+2\frac{\partial ^2f}{\partial u\partial v}+\frac{\partial ^2f}{\partial v^2}.$$ Now, just subtract the first from the second. Note that, as you did, I'm omitting the writing the arguments of the functions, but these are important when applying the chain rule.
In general, if $u(x(\xi,\eta),y(\xi,\eta))=U(\xi,\eta),$ then $$u_{xx}=U_{\xi\xi}(\xi_x)^2+2U_{\xi\eta}\xi_x\eta_x+U_{\eta\eta}(\eta_x)^2+U_\xi\xi_{xx}+U_\eta \eta_{xx}.$$