(*)Let $$Q(h)=\sum^n_{i,j=1}\partial_i\partial_jf(x_0)h_ih_j$$ be the quadratic form associated with the Hessian matrix $H(x)$ of $f$ at $x_0$ with $f:U\subseteq R^n\rightarrow R$ for which the second derivative is a continuous function and $x_0\in U$. (and I think, as it wasn't explicit) that $h$ is the vector you differentiate by (the basis vectors in this case x,y ).
Can someone explain what exactly is being differentiated in this equation? Do you first differentiate the general function $f$ and then evaluate that derivative at $x_0$ or do you multiply the function with the components of $h$? Generally: if you have $\partial_1\partial_2$ do you differentiate the whole expression after it if there are no parentheses or just the function?
(*) I translated this from another language so sorry if the terminology isn't on point.