Could someone please explain the derivation if so?
Kepler's 3 laws are descriptions of how 2 bodies move if the force between them is an inverse square attraction.
He derived them from experimental observations of planets and so they are 'correct'. Newton later calculated that an inverse square law would explain these properties of orbits and so decided that gravity followed this equation.
So except that the only apply to a 2 body system (they don't take into account the effect of other planets) they agree with Newton's law of gravitation.
For two bodies, Newton's laws and Kepler's laws agree if and only if the mass of one body (the "planet") is negligible compared with the mass of the other (the "sun"), so that the latter can be considered stationary. I've written an elementary derivation of Kepler's laws from Newton's laws for that special case. On the relationship between that special case and the general two-body problem, see e.g. Peraire & Widnall.