The use of change of variables and the chain rule 
We introduce new variables as 
  $\begin{cases}\xi:=x+ct\\\eta:=x-ct\end{cases}
$
  which implies that
  $
\begin{cases} \partial_ x=\partial_\xi+\partial_\eta\\\partial_t=c\partial_\xi+c\partial_\eta
\end{cases}
$

This is from page 34 of Partial Differential Equation --- an Introduction (2nd edition) by Strauss. I don't understand the implication here. 
How does this proceeds? Thanks a lot.

There is a typo in Strauss's book. One should have
$$
\partial_t=c\partial_\xi-c\partial_\eta.
$$
Note that both $\xi$ and $\eta$ are functions of $x$ and $t$:
$$
\xi=h(x,t);\quad \eta=g(x,t).
$$
Now, suppose you have a function $u=u(\xi,\eta)$. Then you have
$$
u=u(h(x,t),g(x,t)).
$$
Can you write down by chain rule, what is $u_x$ and $u_t$?
