I apologize for the title, if you can come up with a better one please do it.
In my PDE textbook, it defines $u(x,y,t)=X(x)Y(y)T(t)$ and then seems to claim that the laplacian of u is $X’’YT+XY’’T$. I’m very confused as to why this is the case, as I would expect the laplacian to be
$X’’YT+XY’’T+XYT’’$. This is not a small detail either, as the example only works if the laplacian is what they said it should be. 
In the heat equation $u_t = k\Delta u$ and in the wave equation $u_{tt}=c^2\Delta u$, the Laplacian operator $\Delta$ is taken with respect to space variables only; time derivative is not included in it. This is why, when $u$ is separated as $XYT$, the Laplacian ends up being $(X''Y+XY'')T$.