Integrating Charge Conservation Equation with Dirac Delta Function

[jmwilli25]

Homework Statement

I just need help integrating this equation.
e is just the charge of an electron so it is constant

Homework Equations

$$-\int_{V}d\vec{r}\frac{\delta}{\delta t}e\delta(\vec{r}-\vec{R}(t))$$

The Attempt at a Solution

$$-e\frac{d}{dt}\int_{V}d\vec{r}\delta(\vec{r}-\vec{R}(t))$$
$$-e\frac{d}{dt}4\pi$$

I don't believe my solution is correct or complete

 

[Mark44]

Some additional context would be helpful/necessary. Is the derivative an ordinary derivative like this?
$$-\int_{V}d\vec{r}\frac{d}{dt}e\delta(\vec{r}-\vec{R}(t))$$

or is a partial derivative, like this?
$$-\int_{V}d\vec{r}\frac{\partial}{\partial t}e\delta(\vec{r}-\vec{R}(t))$$

Is $\delta$ just a constant, or are you indicating an impulse function?

 

[hunt_mat]

I think he maybe referring to functional differentiation, not too sure though.

 

[jmwilli25]

Sorry, it is the partial derivative inside the integral and
$$\delta(\vec{r}-\vec{R}(t))$$
is the dirac delta function.

Context: I am trying to show that the equation of charge conservation holds when
$$\rho(\vec{r},t)=e\delta(\vec{r}-\vec{R}(t))$$

The entire equation that I am trying to solve is
$$-\int_{V}d\vec{r}e\frac{\partial}{\partial t}e\delta(\vec{r}-\vec{R}(t))=\int_{V}d\vec{r}\vec{\nabla}\bullet(e\frac{d}{dt}\delta(\vec{r}-\vec{R}(t))).$$
I have to show that the two sides are equal.
But I figured if I could get help just with the LHS then I might be able to do the RHS myself. Granted, I still need to figure out how to apply the divergence to a full derivative of time, but one step at a time. Also, the equation that I have just given is exactly what is in the book. Classical Electordynamics, Schwinger 1998.