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Explicit expression for the derivative

We can rewrite

\nabla_{\boldsymbol{\Theta}}\log{Z(\boldsymbol{\Theta})}=\frac{\nabla_{\boldsymbol{\Theta}}Z(\boldsymbol{\Theta})}{Z(\boldsymbol{\Theta})},

which reads in more detail

\nabla_{\boldsymbol{\Theta}}\log{Z(\boldsymbol{\Theta})}=\frac{\nabla_{\boldsymbol{\Theta}} \sum_{x_i\in \boldsymbol{X}}f(x_i;\boldsymbol{\Theta}) }{Z(\boldsymbol{\Theta})}.

We can rewrite the function f (we have assumed that is larger or equal than zero) as f=\exp{\log{f}} . We can then reqrite the last equation as

\nabla_{\boldsymbol{\Theta}}\log{Z(\boldsymbol{\Theta})}=\frac{ \sum_{x_i\in \boldsymbol{X}} \nabla_{\boldsymbol{\Theta}}\exp{\log{f(x_i;\boldsymbol{\Theta})}} }{Z(\boldsymbol{\Theta})}.