For research purposes, I am solving a particular case of an optimal control problem. I need to compute $$\nabla_{u_t} (x_t^TC^TQ_tv_t) $$ where $x_t \in \mathbb{R}^n$, $ C \in \mathbb{R}^{q \times n}$ and $Q_t \in \mathbb{R}^{q \times q}$ do not depend on $u_t \in \mathbb{R}^m$, while the noise $v_t \in \mathbb{R}^q$ is $$ v_t \sim N(0,V_t(u_t)),$$ with covariance matrix $V_t$ which depends on $u_t$. The suffix $T$ denotes the transpose operator.
Does anyone know how to compute this?