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[itex] \frac{d}{d x_1} \left[ f(x_1, x_2, x_3) \right] [/itex]

where [itex] x_1 = g(x_2, x_3) [/itex]

studying thermodynamics right now, i'm encountering into expressions like

[itex] \mu = - \left(\frac{\partial U}{\partial S} \right)_{N,V} \left( \frac{\partial S}{\partial N} \right)_{U,V} [/itex]

where some particular variables are held fixed, but others are not. i'm wondering if there are formal theorems relating to partial derivatives like these or if you guys have any knowledge specifically relating to them.