汎函数微分
{\displaystyle {\begin{aligned}\left\langle {\frac {\delta H}{\delta p}},\phi \right\rangle &=\sum _{x}{\frac {\delta H[p(x)]}{\delta p(x')}}\,\phi (x')\\&=\left.{\frac {d}{d\epsilon }}H[p(x)+\epsilon \phi (x)]\right|_{\epsilon =0}\\&=-{\frac {d}{d\varepsilon }}\left.\sum _{x}[p(x)+\varepsilon \phi (x)]\log[p(x)+\varepsilon \phi (x)]\right|_{\varepsilon =0}\\&=\displaystyle -\sum _{x}[1+\log p(x)]\phi (x)\\&=\left\langle -[1+\log p(x)],\phi \right\rangle .\end{aligned}}}
即ち、
δ H δ p = − [ 1 + log  p ( x ) ] {\displaystyle {\frac {\delta H}{\delta p}}=-[1+\log p(x)]}
が成り立つ。