Interpretation of Raoult law
The partial pressure of a component in a mixture is always less than that of the pure component at the same temperature. (Calculating the chemical potential shows that if it were not the case the mixture would be unstable compared to the pure liquid).  The question then becomes what is the partial pressure of the component A above a mixture $p_A$ related to that above the pure liquid $p_A^*$.  Raoult's law states that the ratio is equal to the mole fraction $x_A$ of A, $p_A/p_A^*=x_A$. 

Physically, this is justified by considering that the chance of finding a molecule of A in the vapour, relative to the chance for the pure liquid, is equal to its mole fraction. This is reasonable if dealing with ideal mixtures and assuming that the vapour behaves as an ideal gas. Thus there is no derivation as such, we proceed directly from the physical argument to the law.