Why isn't the product rule used in the definition of mechanical work?

Question

Mechanical power is normally defined as $P = \mathrm{d}W/\mathrm{d}t$, and work is normally defined as $W = \vec F \cdot \vec x$. Today an undergrad pointed out a confusion he had from Griffiths' E&M book where he has the line

$$P = \frac{\mathrm{d}W}{\mathrm{d}t} = \vec F \cdot \vec v$$

Which is a definition of power I had seen before. But the student was confused because it seems like, if you have a time dependent force $\vec F(t)$, the math should work out like:

$$P = \frac{\mathrm{d}W}{\mathrm{d}t} = \frac{\mathrm{d}\vec F(t)}{\mathrm{d}t} \cdot \vec x + \vec F \cdot \frac{\mathrm{d}\vec x(t)}{\mathrm{d}t}$$

(From the product rule.)

However, I've never seen the above formula, so I'm guessing it's wrong. Also, it clashes with the $\vec F \cdot \vec v$ version that I'm pretty sure works completely fine with a time dependent force.

I gave him a pretty weak answer and warned him that it's probably not correct: I told him that if you start with the differential form of work, $dW = \vec F \cdot d \vec x$, it seems to assume that the force stays the same in that tiny $d\vec x$, so it's constant there, and then dividing both by $dt$ gives you the equation we're looking for.

Wikipedia seems to say something similar, basically using the step of $d\vec x = \vec v \ dt$ to get $\vec F \cdot \vec v$, which also assumes that $\vec v$ is constant over $d \vec x$.

So, assuming one of those are correct, I see how they get $\vec F \cdot \vec v$. But what's the flaw in the product rule thing?

Answer 1

The error in the original reasoning comes from asserting that $W=\vec F\cdot \vec x$. This is only true if the force is constant.

For a particle traveling along a parameterized curve $\vec x(t)$ and under the influence of a force $\vec F(\vec x,t)$ which is explicitly dependent on both position in space and time, the work performed on the particle by this force from a time $t_0$ to a time $t$ is defined as follows: \begin{align} W_{t_0}(t) = \int_{t_0}^{t} \vec F(\vec x(t'),t')\cdot \dot{\vec x}(t') \,dt' \end{align} Note that the expression on the right is often written $\int \vec F\cdot \vec dx$, but this is really schematic, the the mathematically precise definition is what I have written above in terms of a parameterized path with an integral over some range of parameter values. The definition of instantaneous power is then \begin{align} P(t) = \dot W_{t_0}(t) \end{align} Taking the derivative of both sides with respect to $t$, and using the fundamental theorem of calculus, we obtain the desired expression for the power \begin{align} P(t) = F(\vec x(t),t)\cdot \dot{\vec x}(t) \end{align}