When the conservative force F\left(x\right) (for simplicity, in one dimension) does work W on a particle within the system, the change in potential energy \triangle U of the system is equal to the negative of the work done by the conservative force, i.e.,
\;\;\;\;\;\;\;\;\;\;\;\triangle U=-WBut, \;\;\;\;\;\;W=\int_{x_i}^{x_f}F\left(x\right)\;dx
\therefore\;\;\;\;\;\;\;\;\triangle U=-\int_{x_i}^{x_f}F\left(x\right)\;dxDifferentiating the above equation, we get
\frac{\operatorname dU\;\left(x\right)}{\operatorname dx}=-F\left(x\right)or, F\left(x\right)=-\frac{\operatorname dU\;\left(x\right)}{\operatorname dx}
Hence potential energy U may be defined as a function whose negative gradient gives the force. Conversely, we may define conservative force as a force which is equal to the negative gradient of the potential energy U.
