Derivation of Maxwell's relations

In thermodynamics one of the maxwell relations is:
[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?