What is the derivative of $U^{\rm T}(t)V(t)U(t)$?

Question

Is this correct ?

$$\frac{{\rm d}~[U^{\rm T}(t)V(t)U(t)]}{{\rm d}t}=U^{\rm T}(t)V(t){\rm d}(U(t))+U^{\rm T}(t)U(t){\rm d}(V(t))+V(t)U(t){\rm d}(U^{\rm T}(t))$$

Please note that $U\in\mathbb{R}^{n\times 1}$ and $V\in\mathbb{R}^{n\times n}$.

Answer 1

We obtain: $(U')^TVU+U^TV'U+U^TVU'$.