acoustic dipole volume integral w/ dirac delta?

Question

I have an acoustic research problem that leads to the following integral formulation: \begin{align} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}p(\mathbf{y},\tau)\frac{\partial}{\partial y_i}\left(n_i \delta(f) |\nabla f|\right) g(\mathbf{x} - \mathbf{y}, t-\tau) d^3\mathbf{y} d\tau, \end{align} where $p$ is a scalar function, $n_i=\nabla f / |\nabla f|$ is the normal vector of my control surface, and $\delta(f)$ is the dirac delta with my level set function, $f$, the zero set of which describes the control surface. $g$ is a numerical green's function. If legal, I would like to use the vector property, \begin{align} \nabla \cdot (\psi \mathbf{v}) = \nabla \psi \cdot \mathbf{v} + \psi \nabla \cdot \mathbf{v}, \end{align} to convert the integrand to a form such as \begin{align} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}p(\mathbf{y},\tau)\left[ \frac{\partial n_i}{\partial y_i} \left(\delta(f) |\nabla f|\right) + n_i\frac{\partial}{\partial y_i} \left(\delta(f) |\nabla f|\right) \right] g(\mathbf{x} - \mathbf{y}, t-\tau) d^3\mathbf{y} d\tau. \end{align} However, since $\delta(f)$ is a distribution, I am not quite sure if
(1) this is a correct manipulation;
(2) how to deal with the second term in the bracket?

I want to convert this to a surface integral to evaluate my boundary sources, and the first term gives me just that. Thanks in advance.