Characterization of the distribution$\nabla\delta(g(x))$?

Question

Let $g\in C^\infty(\mathbb R^n)$ such that $\nabla g(x) \neq 0$ whenever $g(x)=0$. According to Delta function and integrating over level sets?, the distribution $g^*\delta_0 = \delta(g(x))$ is well-defined and satisfies $$\int_{\mathbb R^n} \delta(g(x)) f(x) dx = \int_{g^{-1}(0)} \frac{f(x)}{|\nabla g(x)|} dx,$$ where $f\in C_c^\infty(\mathbb R^n)$. Then, what about the derivative $g^*(\nabla \delta_0) = \nabla\delta(g(x))$? I want an analogous formula of the form $$\int \nabla\delta(g(x)) f(x) dx = \int_{g^{-1}(0)} (???)$$