Question

What is the derivative of the work function?

Answer 1

It depends with respect to what physical quantity you're differentiating.

If you consider the derivative with respect to time, it is the power, by definition:

`P = (dW)/(dt)`

If you consider the derivative of the work with respect to position, we have the following result, using the Fundamental Theorem of Calculus:

`(dW)/(dx) = d/(dx) int_(a)^(x) F(x^prime) dx^prime = F(x)`

Which is the force.

This last result can be generalized to higher dimensions, as long as the force is conservative.