There is a standard definition that $\mathrm{d} f$ means a cotangent vector field (1-form) when $f$ is a differentiable function on $\mathbb{R}^n$ or a smooth $n$-manifold $M$, by which we can make the expression $$ \mathbb{d} f= \sum_{i=1}^n \frac{\partial f}{\partial x_i} \mathrm{d}x_i $$ hold mathematically and rigorously.
My question is: can we do a similar thing to $\mathrm{d} B_t$ (or $\mathrm{d} W_t$) in an Ito integral so that it is also a well-defined mathematical rigorous object, and Ito formula $$ d X_t=\mu_t d t+\sigma_t d B_t $$
$$ d f(X_t)=\left(\frac{\partial f}{\partial t}+\mu_t \frac{\partial f}{\partial x}+\frac{\sigma_t^2}{2} \frac{\partial^2 f}{\partial x^2}\right) d t+\sigma_t \frac{\partial f}{\partial x} d B_t $$ be rigorously correct?