$$
\begin{align}
    \frac{dln(p(x))}{d\theta} &= ln(\Sigma_{h}e^{-E(x,h)}) - ln(\Sigma_{x}\Sigma_{h}e^{-E(x,h)}) \\
    &= -\frac{1}{\Sigma_{h}e^{-E(x,h)}}
    \Sigma_{h}
    p(x,h)\Sigma_x\Sigma_he^{-E(x,h)}
    \frac{dE(x,h)}{d\theta} +
    \frac{1}{\Sigma_x\Sigma_he^{-E(x,h)}}
    \Sigma_x\Sigma_xe^{-E(x,h)}\frac{dE(x,h)}{d\theta}
    \\
    &= -\frac{1}{\Sigma_{h}p(x,h)\Sigma_x\Sigma_he^{-E(x,h)}}
    \Sigma_{h}
    e^{-E(x,h)}
    \frac{dE(x,h)}{d\theta} +
    \frac{1}{\Sigma_x\Sigma_he^{-E(x,h)}}
    \Sigma_x\Sigma_hp(x,h)\Sigma_x\Sigma_he^{-E(x,h)}\frac{dE(x,h)}{d\theta}
    \\
    &= -\frac{1}{\Sigma_{h}p(x,h)}
    \Sigma_{h}
    (p(x,h)\frac{dE(x,h)}{d\theta}) +
    \Sigma_x\Sigma_hp(x,h)\frac{dE(x,h)}{d\theta}
    \\
    &= -\Sigma_{h}
    (\frac{p(x,h)}{p(v)}\frac{dE(x,h)}{d\theta}) +
    \Sigma_x\Sigma_hp(x,h)\frac{dE(x,h)}{d\theta}
    \\
    &= -\Sigma_{h}
    (p(h|x)\frac{dE(x,h)}{d\theta}) +
    \Sigma_x\Sigma_hp(x,h)\frac{dE(x,h)}{d\theta}
    \\
    &= E_h(\frac{dE(x,h)}{d\theta}|x) - E_{h,x}[\frac{dE(x,h)}{d\theta}]
\end{align}
$$