Rowan Bray

2023-03-27

What is the derivative of the work function?

pistuiescvzeb

Beginner2023-03-28Added 10 answers

It is dependent on the physical quantity being differentiated.

When considering the derivative with respect to time, it is, by definition, the power:

$P=\frac{dW}{dt}$

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

$\frac{dW}{dx}=\frac{d}{dx}{\int}_{a}^{x}F\left({x}^{\prime}\right){dx}^{\prime}=F\left(x\right)$

Which is the force.

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

When considering the derivative with respect to time, it is, by definition, the power:

$P=\frac{dW}{dt}$

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

$\frac{dW}{dx}=\frac{d}{dx}{\int}_{a}^{x}F\left({x}^{\prime}\right){dx}^{\prime}=F\left(x\right)$

Which is the force.

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

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