# Deriving PV^n=constant for polytropic processes

Deriving $P{V}^{n}=\text{constant}$ for polytropic processes?
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bardalhg
The dU here is an exact differential:
$\mathrm{d}U\left(T,\phantom{\rule{thinmathspace}{0ex}}V\right)={\left(\frac{\mathrm{\partial }U}{\mathrm{\partial }T}\right)}_{V}\mathrm{d}T+{\left(\frac{\mathrm{\partial }U}{\mathrm{\partial }V}\right)}_{T}\mathrm{d}V$
rather than the derivatives you have. Hence, you cannot cancel the dT's and dV's as you've done.