Ito formula for a function of class $C^1$

Question

Can the Ito formula be applied with a $C^1$ function if the second order terms vanish ? For example, let $g(t)$ be a function of class $C^1$ and define $F(x,t)=xg(t)$ which is also of class $C^1$. Denote by $B_t$ the standard Browmian motion. Then, the explicit Ito decomposition of $F(B_t,t)$ would be : $F(B_t,t) = \int_0^tg(s)dB_s + \int_0^tg'(s)B_sds$. There are no second order terms because the only one with non-zero cross-quadratic variation is
$\frac{1}{2}\int_0^t\frac{\partial^2F}{\partial x^2}(B_s,s)ds$ and $\frac{\partial^2F}{\partial x^2}=0$. Is this justified ?