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[itex]

\left( \frac{\partial S}{\partial V} \right)_T = \left( \frac{\partial P}{\partial T} \right)_V

[/itex]

When I try to derive it from [itex] dU = TdS - PdV [/itex] i get:

[itex]

T = \left( \frac{\partial U}{\partial S} \right)_V

[/itex]

[itex]

P = -\left( \frac{\partial U}{\partial V} \right)_S

[/itex]

[itex]

\left( \frac{\partial T}{\partial V} \right)_S = \frac{\partial}{\partial V}\left( \frac{\partial U}{\partial S} \right)_V = \frac{\partial}{\partial S}\left( \frac{\partial U}{\partial V}\right)_S = -\left( \frac{\partial P}{\partial S} \right)_V

[/itex]

I then multiply with [itex]\frac{\partial S}{\partial T}[/itex],

[itex]

\frac{\partial S}{\partial T} \left( \frac{\partial T}{\partial V} \right)_S = \frac{\partial S}{\partial T} \left( -\frac{\partial P}{\partial S} \right)_V

[/itex]

[itex]

\left( \frac{\partial S}{\partial V} \right)_S = -\left( \frac{\partial P}{\partial T} \right)_V

[/itex]

So, what am I doing wrong?