Prove $F = - \frac{dV}{dx}$ for a conservative force.

(N/A) Suppose that a body undergoes a small displacement $dx$ under the action of a conservative force $F$.
The work done $dW$ by this conservative force is given by $dW = F dx$.
According to the work-energy theorem,the work done by all forces is equal to the change in kinetic energy,$dW = dK$.
From the law of conservation of mechanical energy,the total mechanical energy $E = K + V$ remains constant for a conservative force,meaning $dE = 0$.
Therefore,$dK + dV = 0$,which implies $dK = -dV$.
Substituting $dW = dK$ and $dW = F dx$,we get $F dx = -dV$.
Thus,$F = -\frac{dV}{dx}$.
Hence,for a conservative force,the force is equal to the negative gradient of the potential energy with respect to displacement.