I think this is a fairly basic question but I am just unable to crack it.
Given that $ u= f(x) + g(y)$, how do I write $\frac{\partial^2 \psi}{\partial u^2}$ in terms of $ \frac{\partial^2 \psi}{\partial x^2}$ and $\frac{\partial^2 \psi}{\partial y^2}$ for some $\psi(u)?$?
I think this involves the chain rule but I'm not sure how to apply it for partial derivatives. Any help would be really appreciated. Thank you.
Hint:
If $u(x,y)=f(x)+g(y)$ and $x,y$ are independent variable than $$ \frac{\partial u}{\partial x}=f'(x) \qquad\frac{\partial u}{\partial y}=g'(y) $$ and
$$ \frac{\partial^2 u}{\partial x^2}=\frac{\partial}{\partial x}f'(x)=f''(x) \qquad\frac{\partial^2 u}{\partial y^2}=\frac{\partial}{\partial y}g'(y)=g''(y) $$
Now use the chain rule to find the derivatives of $\psi(u)$