We have:
$\triangle f + \lambda f = 0$ on $D$
$f = 0$ on $\partial D$
For some domain $D$ with smooth boundary. $\lambda$ is the corresponding eigenvalue to the eigenvector $f$. $\triangle$ being the Laplace operator: $\triangle f = f_{xx}+f_{yy}$. And I have to show that the eigenfunctions for different eigenvalues are orthogonal with respect to the inner product: $$ \left \langle f,g \right \rangle = \iint_{D}^{} fg dx dy $$
I am given a hint to use Green's formula together with product rule for derivatives to "move derivatives on factors in integrands". I think that ultimately I want, using Green's formula, every integral to be on $\partial D$, since that would yield 0. The first thing to do is to write out $f$ and $g$. I'm going to let $\lambda _f$ be the eigenvalue for the eigenvector $f$ and $\lambda _g$ be the eigenvalue for the eigenvector $g$. This makes the inner product:
$$ \frac{1}{\lambda _f \lambda _g}\iint_{D}^{} (f_{xx}+f_{yy})(g_{xx}+g_{yy})dxdy $$
And I want this to be 0 then. I've tried multiple times but I always get stuck somewhere. What I'm wondering then is how I can think? Or if someone can spell out what the hint is trying to tell me, I haven't worked out how to use the product rule for derivatives in my favour.