How to Rigorously Derive P(t) = V(t)I(t) in an Electric Circuit?

[a4b3c2d1e0f]

I kindly ask for assistance in derivation of the equation for instantaneous power in an electric circuit, P(t) = V(t) I(t). I want to derive it as rigorously as possible. Here's what I got:
We start with $P = {\bf F} \cdot {\bf v}$, where ${\bf v} = \frac{d\bf r}{dt}$
We know that the force exerted on a test charge q is given by ${\bf F} = {\bf E} q$, and for voltage we know $dV = - {\bf E} \cdot dx$.
Inserting F in equation for power, we get $P = {\bf E}q \cdot \frac{d\bf x}{dt} = {\bf E}\cdot dx \frac{q}{dt} = - dV \frac{q}{dt} .$
How would I go from this, to the desired result, without taking a "quantum leap"?
Is there a better way to actually derive this mathematically impeccably?

 

[Studiot]

You need first to decide whether you want the power in a circuit, as you described,

or the flow of power through a single point in space, which you appear to be trying to calculate.

The old fashioned definition of EMF made this quite clear for a circuit.

The EMF of a circuit develops the power in that circuit, equal to the EMF times the current flowing in that circuit.