The partition function is the generating function of expectation values, in particular there are mathematical relationships between expectation values and the log-partition function. In this case we have
$$ \begin{align*} \langle \frac{ \partial E(\boldsymbol{x}; \Theta_i) } { \partial \Theta_i} \rangle_{model} = \int p(\boldsymbol{x}| \boldsymbol{\Theta}) \frac{ \partial E(\boldsymbol{x}; \Theta_i) } { \partial \Theta_i} d\boldsymbol{x} = -\frac{\partial \log Z(\Theta_i)}{ \partial \Theta_i} . \end{align*} $$Here \( \langle \cdot \rangle_{model} \) is the expectation value over the model probability distribution \( p(\boldsymbol{x}| \boldsymbol{\Theta}) \).