求导函数：
输入一个原函数（即需要求导的函数），然后设置需要求导的变量和求导的阶数，点击“下一步”按钮，即可获得该函数相应阶数的导函数。
注意，输入的函数支持数学函数和其它常量。
输入一个原函数（即需要求导的函数），然后设置需要求导的变量和求导的阶数，点击“下一步”按钮，即可获得该函数相应阶数的导函数。
注意，输入的函数支持数学函数和其它常量。
本次共计算 1 个题目：每一题对 T 求 2 阶导数。
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数-kTln(\frac{1}{2}(1 + cosh(\frac{j}{(kT)}) + {(8 + {((2cosh(\frac{j}{(kT)})) - 1)}^{2})}^{\frac{1}{2}})) 关于 T 的 2 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = -kTln(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( -kTln(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})\right)}{dT}\\=&-kln(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2}) - \frac{kT(\frac{\frac{1}{2}sinh(\frac{j}{kT})j*-1}{kT^{2}} + \frac{1}{2}(\frac{\frac{1}{2}(\frac{4*2cosh(\frac{j}{kT})sinh(\frac{j}{kT})j*-1}{kT^{2}} - \frac{4sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}}) + 0)}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})}\\=&-kln(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2}) + \frac{jsinh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T} + \frac{2jsinh(\frac{j}{kT})cosh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T} - \frac{jsinh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( -kln(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2}) + \frac{jsinh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T} + \frac{2jsinh(\frac{j}{kT})cosh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T} - \frac{jsinh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T}\right)}{dT}\\=&\frac{-k(\frac{\frac{1}{2}sinh(\frac{j}{kT})j*-1}{kT^{2}} + \frac{1}{2}(\frac{\frac{1}{2}(\frac{4*2cosh(\frac{j}{kT})sinh(\frac{j}{kT})j*-1}{kT^{2}} - \frac{4sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}}) + 0)}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})} + \frac{(\frac{-(\frac{\frac{1}{2}sinh(\frac{j}{kT})j*-1}{kT^{2}} + \frac{1}{2}(\frac{\frac{1}{2}(\frac{4*2cosh(\frac{j}{kT})sinh(\frac{j}{kT})j*-1}{kT^{2}} - \frac{4sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}}) + 0)}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})^{2}})jsinh(\frac{j}{kT})}{2T} + \frac{j*-sinh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T^{2}} + \frac{jcosh(\frac{j}{kT})j*-1}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})TkT^{2}} + \frac{2(\frac{\frac{-1}{2}(\frac{4*2cosh(\frac{j}{kT})sinh(\frac{j}{kT})j*-1}{kT^{2}} - \frac{4sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{3}{2}}})jsinh(\frac{j}{kT})cosh(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T} + \frac{2(\frac{-(\frac{\frac{1}{2}sinh(\frac{j}{kT})j*-1}{kT^{2}} + \frac{1}{2}(\frac{\frac{1}{2}(\frac{4*2cosh(\frac{j}{kT})sinh(\frac{j}{kT})j*-1}{kT^{2}} - \frac{4sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}}) + 0)}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})^{2}})jsinh(\frac{j}{kT})cosh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}T} + \frac{2j*-sinh(\frac{j}{kT})cosh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T^{2}} + \frac{2jcosh(\frac{j}{kT})j*-cosh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})TkT^{2}} + \frac{2jsinh(\frac{j}{kT})sinh(\frac{j}{kT})j*-1}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})TkT^{2}} - \frac{(\frac{\frac{-1}{2}(\frac{4*2cosh(\frac{j}{kT})sinh(\frac{j}{kT})j*-1}{kT^{2}} - \frac{4sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{3}{2}}})jsinh(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T} - \frac{(\frac{-(\frac{\frac{1}{2}sinh(\frac{j}{kT})j*-1}{kT^{2}} + \frac{1}{2}(\frac{\frac{1}{2}(\frac{4*2cosh(\frac{j}{kT})sinh(\frac{j}{kT})j*-1}{kT^{2}} - \frac{4sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}}) + 0)}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})^{2}})jsinh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}T} - \frac{j*-sinh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T^{2}} - \frac{jcosh(\frac{j}{kT})j*-1}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})TkT^{2}}\\=&\frac{2jsinh(\frac{j}{kT})cosh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T^{2}} - \frac{2jsinh(\frac{j}{kT})cosh(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}T^{2}} + \frac{8j^{2}sinh^{2}(\frac{j}{kT})cosh^{2}(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{3}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})kT^{3}} + \frac{2j^{2}sinh^{2}(\frac{j}{kT})cosh(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})^{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}kT^{3}} - \frac{8j^{2}sinh^{2}(\frac{j}{kT})cosh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{3}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})kT^{3}} - \frac{j^{2}cosh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})kT^{3}} + \frac{4j^{2}sinh^{2}(\frac{j}{kT})cosh^{2}(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})^{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)kT^{3}} - \frac{4j^{2}sinh^{2}(\frac{j}{kT})cosh(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})^{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)kT^{3}} - \frac{j^{2}sinh^{2}(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})^{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}kT^{3}} - \frac{jsinh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T^{2}} - \frac{2j^{2}cosh^{2}(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}kT^{3}} - \frac{2j^{2}sinh^{2}(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}kT^{3}} + \frac{j^{2}sinh^{2}(\frac{j}{kT})}{4(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})^{2}kT^{3}} + \frac{2j^{2}sinh^{2}(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{3}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})kT^{3}} + \frac{j^{2}sinh^{2}(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})^{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)kT^{3}} + \frac{jsinh(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}T^{2}} + \frac{j^{2}cosh(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}kT^{3}}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数-kTln(\frac{1}{2}(1 + cosh(\frac{j}{(kT)}) + {(8 + {((2cosh(\frac{j}{(kT)})) - 1)}^{2})}^{\frac{1}{2}})) 关于 T 的 2 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = -kTln(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( -kTln(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})\right)}{dT}\\=&-kln(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2}) - \frac{kT(\frac{\frac{1}{2}sinh(\frac{j}{kT})j*-1}{kT^{2}} + \frac{1}{2}(\frac{\frac{1}{2}(\frac{4*2cosh(\frac{j}{kT})sinh(\frac{j}{kT})j*-1}{kT^{2}} - \frac{4sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}}) + 0)}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})}\\=&-kln(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2}) + \frac{jsinh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T} + \frac{2jsinh(\frac{j}{kT})cosh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T} - \frac{jsinh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( -kln(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2}) + \frac{jsinh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T} + \frac{2jsinh(\frac{j}{kT})cosh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T} - \frac{jsinh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T}\right)}{dT}\\=&\frac{-k(\frac{\frac{1}{2}sinh(\frac{j}{kT})j*-1}{kT^{2}} + \frac{1}{2}(\frac{\frac{1}{2}(\frac{4*2cosh(\frac{j}{kT})sinh(\frac{j}{kT})j*-1}{kT^{2}} - \frac{4sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}}) + 0)}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})} + \frac{(\frac{-(\frac{\frac{1}{2}sinh(\frac{j}{kT})j*-1}{kT^{2}} + \frac{1}{2}(\frac{\frac{1}{2}(\frac{4*2cosh(\frac{j}{kT})sinh(\frac{j}{kT})j*-1}{kT^{2}} - \frac{4sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}}) + 0)}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})^{2}})jsinh(\frac{j}{kT})}{2T} + \frac{j*-sinh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T^{2}} + \frac{jcosh(\frac{j}{kT})j*-1}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})TkT^{2}} + \frac{2(\frac{\frac{-1}{2}(\frac{4*2cosh(\frac{j}{kT})sinh(\frac{j}{kT})j*-1}{kT^{2}} - \frac{4sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{3}{2}}})jsinh(\frac{j}{kT})cosh(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T} + \frac{2(\frac{-(\frac{\frac{1}{2}sinh(\frac{j}{kT})j*-1}{kT^{2}} + \frac{1}{2}(\frac{\frac{1}{2}(\frac{4*2cosh(\frac{j}{kT})sinh(\frac{j}{kT})j*-1}{kT^{2}} - \frac{4sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}}) + 0)}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})^{2}})jsinh(\frac{j}{kT})cosh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}T} + \frac{2j*-sinh(\frac{j}{kT})cosh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T^{2}} + \frac{2jcosh(\frac{j}{kT})j*-cosh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})TkT^{2}} + \frac{2jsinh(\frac{j}{kT})sinh(\frac{j}{kT})j*-1}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})TkT^{2}} - \frac{(\frac{\frac{-1}{2}(\frac{4*2cosh(\frac{j}{kT})sinh(\frac{j}{kT})j*-1}{kT^{2}} - \frac{4sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{3}{2}}})jsinh(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T} - \frac{(\frac{-(\frac{\frac{1}{2}sinh(\frac{j}{kT})j*-1}{kT^{2}} + \frac{1}{2}(\frac{\frac{1}{2}(\frac{4*2cosh(\frac{j}{kT})sinh(\frac{j}{kT})j*-1}{kT^{2}} - \frac{4sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}}) + 0)}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})^{2}})jsinh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}T} - \frac{j*-sinh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T^{2}} - \frac{jcosh(\frac{j}{kT})j*-1}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})TkT^{2}}\\=&\frac{2jsinh(\frac{j}{kT})cosh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T^{2}} - \frac{2jsinh(\frac{j}{kT})cosh(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}T^{2}} + \frac{8j^{2}sinh^{2}(\frac{j}{kT})cosh^{2}(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{3}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})kT^{3}} + \frac{2j^{2}sinh^{2}(\frac{j}{kT})cosh(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})^{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}kT^{3}} - \frac{8j^{2}sinh^{2}(\frac{j}{kT})cosh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{3}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})kT^{3}} - \frac{j^{2}cosh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})kT^{3}} + \frac{4j^{2}sinh^{2}(\frac{j}{kT})cosh^{2}(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})^{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)kT^{3}} - \frac{4j^{2}sinh^{2}(\frac{j}{kT})cosh(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})^{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)kT^{3}} - \frac{j^{2}sinh^{2}(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})^{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}kT^{3}} - \frac{jsinh(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})T^{2}} - \frac{2j^{2}cosh^{2}(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}kT^{3}} - \frac{2j^{2}sinh^{2}(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}kT^{3}} + \frac{j^{2}sinh^{2}(\frac{j}{kT})}{4(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})^{2}kT^{3}} + \frac{2j^{2}sinh^{2}(\frac{j}{kT})}{(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{3}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})kT^{3}} + \frac{j^{2}sinh^{2}(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})^{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)kT^{3}} + \frac{jsinh(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}T^{2}} + \frac{j^{2}cosh(\frac{j}{kT})}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}} + \frac{1}{2})(4cosh^{2}(\frac{j}{kT}) - 4cosh(\frac{j}{kT}) + 9)^{\frac{1}{2}}kT^{3}}\\ \end{split}\end{equation} \]
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