What is the derivative of the work function?

1 Answer

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.