= Solution
Differentiate the Euler relation for a single-component <simple compressible system>:
$$
dU=T\,dS+S\,dT-p\,dV-V\,dp+\mu\,dN+N\,d\mu.
$$
The thermodynamic fundamental relation for the same system is $dU=T\,dS-p\,dV+\mu\,dN$. Subtraction gives
$$
S\,dT-V\,dp+N\,d\mu=0.
$$
Dividing by $N$, with $s=S/N$ and $v=V/N$, proves the <Gibbs-Duhem equation>
$$
\boxed{d\mu=-s\,dT+v\,dp}.
$$
The underlying Euler relation assumes extensive bulk thermodynamics and no additional work coordinates. The result says that the intensive quantities are constrained; the <chemical potential> cannot be varied independently of <temperature> and <pressure> within one homogeneous phase.
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