Energy conservation follows from Newtons' second law. As an example, consider a particle moving in a gravitational field along the s-axis that is inclined at an angle θ with respect to the horizontal:

Newtons' 2nd law:   F_s = m a_s = m dv_s/dt.

Chain rule of calculus:  m a_s = m dv_s/dt = m (dv_s/ds) (ds/dt) = m v_s (dv_s/ds).

The force is:   F_s = - mg sin θ = - mg dy/ds.

Newton's 2nd law becomes:   m v_s dv_s = - mg dy.

Integrating:   ½ m v_s² + mgh = constant.    Mechanical energy is conserved!

Identify mgh as the gravtational potential energy.