Let $u(x,y)=f(x+y)+g(x-y)$. How can I calculate $\partial u/ \partial x$, $\partial u/ \partial y$, $\partial^2 u/ \partial x^2$, $\partial^2 u/ \partial y^2$ in terms of derivatives of $f$ and $g$.
So I thought of making a new variable such that say, $v=x+y, w=x-y$ and then substituting. But I don’t know where this will lead to. Can someone help me on this path?
The two rules you need are
$$\def\p#1#2{\frac{\partial #1}{\partial #2}}\p{(f+g)}x=\p fx+\p gx$$
and
$$\p{f(g(x,y))}x=f'(g(x,y))\p{g(x,y)}x\;.$$