Solution
= Solution
Expand the increment of the <quadratic form>, using $A^T=A$:
$$
q(v+h)-q(v)=v^TAh+h^TAv+h^TAh=2v^TAh+h^TAh.
$$
The last term obeys $|h^TAh|\leq\|A\|\|h\|^2=o(\|h\|)$. Thus $q$ is differentiable and
$$
\boxed{Dq_v(h)=2v^TAh,\qquad\nabla q(v)=2Av.}
$$
This is the <quadratic form gradient>. The derivative is a linear functional of the increment $h$, whereas the <gradient> is its representing vector under the Euclidean inner product.