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Hi, Everyone! This is the page(first image) from

I want to ask the definition of work(##W=F(x) \Delta x##) by variable force here is somewhat different from the usual integral version. I don't understand how is this valid definition?

Secondly, how did they reach to the derivative F(x)=-dU/dx by limit?

(There is another approach to proving this by taking potential energies and differentiating them which results in the field force involved. This is given in my another book which is to understand(second image)).

**Principle of physics by resnik.**I want to ask the definition of work(##W=F(x) \Delta x##) by variable force here is somewhat different from the usual integral version. I don't understand how is this valid definition?

Secondly, how did they reach to the derivative F(x)=-dU/dx by limit?

(There is another approach to proving this by taking potential energies and differentiating them which results in the field force involved. This is given in my another book which is to understand(second image)).