Let $F: \mathbb{R}^n \to S^n$ be differentiable function at point $x \in \mathbb{R}^n$. Where $S^n$ is the space of all symmetric matrices with usual Euclidian (trace) norm. Assume $v \in \mathbb{R}^n$ be a fix vector. Define the real valued function $f: \mathbb{R}^n \to \mathbb{R}$ with $f(x) = \langle v , F(x) v \rangle$.
My question: Is the any nice representation of $\nabla f(x)$ in terms of derivative of $F$?
As suggested in the comments, the $i^{th}$ component of the gradient $\nabla f(x)$ is just the $i^{th}$ partial derivative $(\partial _if)(x)$, and by the chain rule it can be calculated as \begin{align} (\partial_if)(x) = \left\langle v, (\partial_iF)(x) \cdot v \right\rangle \end{align}
We can also convey the same information in the language of Frechet/total derivatives as follows: $df_x: \Bbb{R}^n \to \Bbb{R}$ is that linear map such that for every $\xi \in \Bbb{R}^n$, \begin{align} df_x(\xi) = \left\langle v, dF_x(\xi) \cdot v \right\rangle \end{align}