= Solution
Write $\dot x=dx/du$, $\ddot x=d^2x/du^2$, $A_v=dA/dv$, $A_{vv}=d^2A/dv^2$, and $y_w=dy/dw$, $y_{ww}=d^2y/dw^2$. Put $S=A+A^T$. The <product rule> and <chain rule> give
$$
\boxed{\begin{aligned}
f_u&=\dot x^TSx,\\
f_v&=x^TA_vx+w' y_w^Tb,\\
f_{uu}&=\ddot x^TSx+\dot x^TS\dot x,\\
f_{uv}&=\dot x^T(A_v+A_v^T)x,\\
f_{vv}&=x^TA_{vv}x+\bigl((w')^2y_{ww}+w''y_w\bigr)^Tb.
\end{aligned}}
$$
Here $w',w''$ are derivatives with respect to $v$. In the first derivative, $x^TA\dot x=\dot x^TA^Tx$ because the expression is scalar. Likewise $\dot x^TS\dot x=2\dot x^TA\dot x$ if desired. This <differentiation of a parameter-dependent quadratic form> does not assume that $A$ is symmetric. The positive linear term $y^Tb$ contributes the positive chain-rule terms in the $v$ derivatives; it is independent of $u$.
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