Let $d\in\mathbb N$, $M$ be a bounded $d$-dimensional properly embedded $C^2$-submanifold$^1$ of $\mathbb R^d$ with boundary, $\nu_{\partial M}$ denote the outward-pointing unit normal field on $\partial M$ and $f:\partial M\to\mathbb R$ be $C^1$-differentiable and $X\in C^1(\mathbb R^d,\mathbb R^d)$.
Are we able to show something like$^2$ $$\int\operatorname{div}_{\partial M}fX\:{\rm d}\sigma_{\partial M}=\int f\langle X,\nu_{\partial M}\rangle\operatorname{div}_{\partial M}\nu_{\partial M}\:{\rm d}\sigma_{\partial M},$$ where $\sigma_{\partial M}$ denotes the surface measure on the Borel $\sigma$-algebra $\mathcal B(\partial M)$? If not, are we able to show this, or something similar, under further assumptions?
$^1$ I guess we need to assume $C^2$-regularity to ensure that $\nu_{\partial M}$ is $C^1$-differentiable.
$^2$ Let $\operatorname P_{\partial M}(a)$ denote the orthogonal projection of $\mathbb R^d$ onto $T_a\:\partial M$ for $a\in\partial M$. If $g:M\to\mathbb R^d$ is $C^1$-differentiable at $a\in\partial M$, then $${\rm D}_{\partial M}g(a):=T_a(f)\circ\operatorname P_{\partial M}(a),$$ where $T_a(f)$ denotes the pushforward of $f$ at $a$, and $$(\operatorname{div}_Mg)(a):=\operatorname{tr}{\rm D}_{\partial M}g(a).$$