How to derive the molar fundamental relation
You have  $$ds=\frac{du}{T}+\frac{P}{T}dv+\left(\frac{u}{T}-s+\frac{Pv}{T}-\frac{\mu}{T}\right)\frac{dN}{N}$$
But, $$\mu=u+Pv-Ts$$
That is, $\mu$ is the Gibbs free energy per mole.  So the term in parenthesis drops out.