Instantaneous power for a variable force

Question

I know that instantaneous power is defined as the time derivative of work done. For a constant, it is easy to prove that this is just the dot product of force and velocity.

However, is Instantaneous power even equal to to F.V for a variable force, and if so, how do I prove it.

I have tried to find this on the internet, but to no avail. I did try differentiating the integral definition of work done by the use of chain ruke and the fundamental theorem of calculus but the answer turned out to be a “regular” product of force and velocity; clearly calculus with vectors is clearly different and I have no experience with it whatsoever.

Answer 1

Given that work is the area under the Force-Distance curve

$$W = \int F(t)\,{\rm d}x$$

and power is the time derivative of work, and for each small-displacement increment ${\rm d}x = v\, {\rm d}t$

$$ P = \tfrac{\rm d}{{\rm d}t} W = \tfrac{\rm d}{{\rm d}t} \int F(t)\,v\,{\rm d}t = F(t)\,v $$