I am trying to follow a derivation in solution chemistry. I'm ok with the derivation to (i.e. the starting point): $x_1d\ln f_1=-x_2d\ln f_2$
Now, the next step presented is: $x_1\left({\partial \ln f_1 \over \partial x_1}\right)_{T,p}=x_2\left({\partial \ln f_2 \over \partial x_2}\right)_{T,p}$
It probably is important to know (this equation is the Gibbs-Duhem equation for a binary mixture) that $x_2=1-x_1$, but otherwise supposedly eqn 2 can be derived from eqn 1.
This seems like it must be really simple, but I am having difficulty following this step. Since $x_2=1-x_1$, I thought that would lead to a negative sign negating the original negative sign in the 1st equation. Not so. I tried partial differentiating the eqn 1 wrt $x_1$ and end up with a mess including partial derivatives of differentials i.e. ${\partial (d\ln f_1) \over \partial x_1}$ which I'm not sure is even defined. I know I must be missing something simple. It almost seems you can write down eqn2 "by inspection" but I am wanting some rigorous understanding of why this is so.