By the two definitions of independence for bivariate continuous RVs: (1) F(x,y)=F_X(x)F_Y(y) and

(2) f(x,y)=f_X(x)f_Y(y).

Prove that these two are equivalent. That is: prove that (1) implies (2) and that (2) implies (1).

I tried to differentiate for one and integrate for the other.