Change of variables and the partial derivative

The second one is correct. Why is the other one wrong? Well when you write $$\frac{\partial u}{\partial \xi}$$what you really are saying is $$\frac{\partial u}{\partial \xi}{\huge|}_\eta$$, i.e. taking the derivative with respect to $\xi$, but keeping $\eta$ fixed. If we do this, we are restricting ourselves to the path $x=-t$, where $t$ varies. But we wanted to represent $$\frac{\partial u}{\partial t}{\huge|}_x$$with a fixed $x$. So this expression fails to express the same quantity. Using the chain rule works to ensure this error does not happen.