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。
\[ \begin{equation}\begin{split}【1/1】求函数N(kT)ln(({(2cosh(\frac{j}{(kT)}))}^{N})(1 + {(tanh(\frac{j}{(kT)}))}^{N})) 关于 T 的 2 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = NkTln((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( NkTln((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})\right)}{dT}\\=&Nkln((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N}) + \frac{NkT(((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))})){tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N}({tanh(\frac{j}{kT})}^{N}((0)ln(tanh(\frac{j}{kT})) + \frac{(N)(\frac{sech^{2}(\frac{j}{kT})j*-1}{kT^{2}})}{(tanh(\frac{j}{kT}))})) + ((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))})))}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})}\\=&Nkln((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N}) - \frac{N^{2}j(2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N}sinh(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Tcosh(\frac{j}{kT})} - \frac{N^{2}j{tanh(\frac{j}{kT})}^{N}(2cosh(\frac{j}{kT}))^{N}sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Ttanh(\frac{j}{kT})} - \frac{N^{2}j(2cosh(\frac{j}{kT}))^{N}sinh(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Tcosh(\frac{j}{kT})}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( Nkln((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N}) - \frac{N^{2}j(2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N}sinh(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Tcosh(\frac{j}{kT})} - \frac{N^{2}j{tanh(\frac{j}{kT})}^{N}(2cosh(\frac{j}{kT}))^{N}sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Ttanh(\frac{j}{kT})} - \frac{N^{2}j(2cosh(\frac{j}{kT}))^{N}sinh(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Tcosh(\frac{j}{kT})}\right)}{dT}\\=&\frac{Nk(((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))})){tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N}({tanh(\frac{j}{kT})}^{N}((0)ln(tanh(\frac{j}{kT})) + \frac{(N)(\frac{sech^{2}(\frac{j}{kT})j*-1}{kT^{2}})}{(tanh(\frac{j}{kT}))})) + ((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))})))}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})} - \frac{(\frac{-(((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))})){tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N}({tanh(\frac{j}{kT})}^{N}((0)ln(tanh(\frac{j}{kT})) + \frac{(N)(\frac{sech^{2}(\frac{j}{kT})j*-1}{kT^{2}})}{(tanh(\frac{j}{kT}))})) + ((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))})))}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})^{2}})N^{2}j(2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N}sinh(\frac{j}{kT})}{Tcosh(\frac{j}{kT})} - \frac{N^{2}j*-(2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N}sinh(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})T^{2}cosh(\frac{j}{kT})} - \frac{N^{2}j((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))})){tanh(\frac{j}{kT})}^{N}sinh(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Tcosh(\frac{j}{kT})} - \frac{N^{2}j(2cosh(\frac{j}{kT}))^{N}({tanh(\frac{j}{kT})}^{N}((0)ln(tanh(\frac{j}{kT})) + \frac{(N)(\frac{sech^{2}(\frac{j}{kT})j*-1}{kT^{2}})}{(tanh(\frac{j}{kT}))}))sinh(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Tcosh(\frac{j}{kT})} - \frac{N^{2}j(2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N}cosh(\frac{j}{kT})j*-1}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})TkT^{2}cosh(\frac{j}{kT})} - \frac{N^{2}j(2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N}sinh(\frac{j}{kT})*-sinh(\frac{j}{kT})j*-1}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Tcosh^{2}(\frac{j}{kT})kT^{2}} - \frac{(\frac{-(((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))})){tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N}({tanh(\frac{j}{kT})}^{N}((0)ln(tanh(\frac{j}{kT})) + \frac{(N)(\frac{sech^{2}(\frac{j}{kT})j*-1}{kT^{2}})}{(tanh(\frac{j}{kT}))})) + ((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))})))}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})^{2}})N^{2}j{tanh(\frac{j}{kT})}^{N}(2cosh(\frac{j}{kT}))^{N}sech^{2}(\frac{j}{kT})}{Ttanh(\frac{j}{kT})} - \frac{N^{2}j*-{tanh(\frac{j}{kT})}^{N}(2cosh(\frac{j}{kT}))^{N}sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})T^{2}tanh(\frac{j}{kT})} - \frac{N^{2}j({tanh(\frac{j}{kT})}^{N}((0)ln(tanh(\frac{j}{kT})) + \frac{(N)(\frac{sech^{2}(\frac{j}{kT})j*-1}{kT^{2}})}{(tanh(\frac{j}{kT}))}))(2cosh(\frac{j}{kT}))^{N}sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Ttanh(\frac{j}{kT})} - \frac{N^{2}j{tanh(\frac{j}{kT})}^{N}((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))}))sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Ttanh(\frac{j}{kT})} - \frac{N^{2}j{tanh(\frac{j}{kT})}^{N}(2cosh(\frac{j}{kT}))^{N}*-sech^{2}(\frac{j}{kT})j*-sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Ttanh^{2}(\frac{j}{kT})kT^{2}} - \frac{N^{2}j{tanh(\frac{j}{kT})}^{N}(2cosh(\frac{j}{kT}))^{N}*-2sech(\frac{j}{kT})sech(\frac{j}{kT})tanh(\frac{j}{kT})j*-1}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Ttanh(\frac{j}{kT})kT^{2}} - \frac{(\frac{-(((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))})){tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N}({tanh(\frac{j}{kT})}^{N}((0)ln(tanh(\frac{j}{kT})) + \frac{(N)(\frac{sech^{2}(\frac{j}{kT})j*-1}{kT^{2}})}{(tanh(\frac{j}{kT}))})) + ((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))})))}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})^{2}})N^{2}j(2cosh(\frac{j}{kT}))^{N}sinh(\frac{j}{kT})}{Tcosh(\frac{j}{kT})} - \frac{N^{2}j*-(2cosh(\frac{j}{kT}))^{N}sinh(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})T^{2}cosh(\frac{j}{kT})} - \frac{N^{2}j((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))}))sinh(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Tcosh(\frac{j}{kT})} - \frac{N^{2}j(2cosh(\frac{j}{kT}))^{N}cosh(\frac{j}{kT})j*-1}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})TkT^{2}cosh(\frac{j}{kT})} - \frac{N^{2}j(2cosh(\frac{j}{kT}))^{N}sinh(\frac{j}{kT})*-sinh(\frac{j}{kT})j*-1}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Tcosh^{2}(\frac{j}{kT})kT^{2}}\\=&\frac{-N^{3}j^{2}(2cosh(\frac{j}{kT}))^{(2N)}{tanh(\frac{j}{kT})}^{(2N)}sinh(\frac{j}{kT})sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})^{2}kT^{3}cosh(\frac{j}{kT})tanh(\frac{j}{kT})} + \frac{N^{3}j^{2}{tanh(\frac{j}{kT})}^{N}(2cosh(\frac{j}{kT}))^{N}sinh(\frac{j}{kT})sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})kT^{3}cosh(\frac{j}{kT})tanh(\frac{j}{kT})} - \frac{N^{3}j^{2}{tanh(\frac{j}{kT})}^{(2N)}(2cosh(\frac{j}{kT}))^{(2N)}sinh(\frac{j}{kT})sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})^{2}kT^{3}cosh(\frac{j}{kT})tanh(\frac{j}{kT})} - \frac{2N^{3}j^{2}(2cosh(\frac{j}{kT}))^{(2N)}{tanh(\frac{j}{kT})}^{N}sinh(\frac{j}{kT})sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})^{2}kT^{3}cosh(\frac{j}{kT})tanh(\frac{j}{kT})} + \frac{N^{3}j^{2}(2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N}sinh(\frac{j}{kT})sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})kT^{3}cosh(\frac{j}{kT})tanh(\frac{j}{kT})} - \frac{N^{2}j^{2}(2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N}sinh^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})kT^{3}cosh^{2}(\frac{j}{kT})} - \frac{N^{2}j^{2}{tanh(\frac{j}{kT})}^{N}(2cosh(\frac{j}{kT}))^{N}sech^{4}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})kT^{3}tanh^{2}(\frac{j}{kT})} - \frac{N^{3}j^{2}(2cosh(\frac{j}{kT}))^{(2N)}{tanh(\frac{j}{kT})}^{N}sinh^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})^{2}kT^{3}cosh^{2}(\frac{j}{kT})} - \frac{N^{3}j^{2}(2cosh(\frac{j}{kT}))^{(2N)}{tanh(\frac{j}{kT})}^{(2N)}sech^{4}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})^{2}kT^{3}tanh^{2}(\frac{j}{kT})} + \frac{N^{3}j^{2}(2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N}sinh^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})kT^{3}cosh^{2}(\frac{j}{kT})} + \frac{N^{3}j^{2}{tanh(\frac{j}{kT})}^{N}(2cosh(\frac{j}{kT}))^{N}sech^{4}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})kT^{3}tanh^{2}(\frac{j}{kT})} - \frac{N^{3}j^{2}{tanh(\frac{j}{kT})}^{(2N)}(2cosh(\frac{j}{kT}))^{(2N)}sinh^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})^{2}kT^{3}cosh^{2}(\frac{j}{kT})} - \frac{2N^{2}j^{2}{tanh(\frac{j}{kT})}^{N}(2cosh(\frac{j}{kT}))^{N}sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})kT^{3}} - \frac{N^{3}j^{2}{tanh(\frac{j}{kT})}^{N}(2cosh(\frac{j}{kT}))^{(2N)}sinh^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})^{2}kT^{3}cosh^{2}(\frac{j}{kT})} + \frac{N^{2}j^{2}(2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N}}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})kT^{3}} - \frac{N^{3}j^{2}(2cosh(\frac{j}{kT}))^{(2N)}sinh^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})^{2}kT^{3}cosh^{2}(\frac{j}{kT})} + \frac{N^{3}j^{2}(2cosh(\frac{j}{kT}))^{N}sinh^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})kT^{3}cosh^{2}(\frac{j}{kT})} - \frac{N^{2}j^{2}(2cosh(\frac{j}{kT}))^{N}sinh^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})kT^{3}cosh^{2}(\frac{j}{kT})} + \frac{N^{2}j^{2}(2cosh(\frac{j}{kT}))^{N}}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})kT^{3}}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数N(kT)ln(({(2cosh(\frac{j}{(kT)}))}^{N})(1 + {(tanh(\frac{j}{(kT)}))}^{N})) 关于 T 的 2 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = NkTln((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( NkTln((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})\right)}{dT}\\=&Nkln((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N}) + \frac{NkT(((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))})){tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N}({tanh(\frac{j}{kT})}^{N}((0)ln(tanh(\frac{j}{kT})) + \frac{(N)(\frac{sech^{2}(\frac{j}{kT})j*-1}{kT^{2}})}{(tanh(\frac{j}{kT}))})) + ((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))})))}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})}\\=&Nkln((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N}) - \frac{N^{2}j(2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N}sinh(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Tcosh(\frac{j}{kT})} - \frac{N^{2}j{tanh(\frac{j}{kT})}^{N}(2cosh(\frac{j}{kT}))^{N}sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Ttanh(\frac{j}{kT})} - \frac{N^{2}j(2cosh(\frac{j}{kT}))^{N}sinh(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Tcosh(\frac{j}{kT})}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( Nkln((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N}) - \frac{N^{2}j(2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N}sinh(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Tcosh(\frac{j}{kT})} - \frac{N^{2}j{tanh(\frac{j}{kT})}^{N}(2cosh(\frac{j}{kT}))^{N}sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Ttanh(\frac{j}{kT})} - \frac{N^{2}j(2cosh(\frac{j}{kT}))^{N}sinh(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Tcosh(\frac{j}{kT})}\right)}{dT}\\=&\frac{Nk(((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))})){tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N}({tanh(\frac{j}{kT})}^{N}((0)ln(tanh(\frac{j}{kT})) + \frac{(N)(\frac{sech^{2}(\frac{j}{kT})j*-1}{kT^{2}})}{(tanh(\frac{j}{kT}))})) + ((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))})))}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})} - \frac{(\frac{-(((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))})){tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N}({tanh(\frac{j}{kT})}^{N}((0)ln(tanh(\frac{j}{kT})) + \frac{(N)(\frac{sech^{2}(\frac{j}{kT})j*-1}{kT^{2}})}{(tanh(\frac{j}{kT}))})) + ((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))})))}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})^{2}})N^{2}j(2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N}sinh(\frac{j}{kT})}{Tcosh(\frac{j}{kT})} - \frac{N^{2}j*-(2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N}sinh(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})T^{2}cosh(\frac{j}{kT})} - \frac{N^{2}j((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))})){tanh(\frac{j}{kT})}^{N}sinh(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Tcosh(\frac{j}{kT})} - \frac{N^{2}j(2cosh(\frac{j}{kT}))^{N}({tanh(\frac{j}{kT})}^{N}((0)ln(tanh(\frac{j}{kT})) + \frac{(N)(\frac{sech^{2}(\frac{j}{kT})j*-1}{kT^{2}})}{(tanh(\frac{j}{kT}))}))sinh(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Tcosh(\frac{j}{kT})} - \frac{N^{2}j(2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N}cosh(\frac{j}{kT})j*-1}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})TkT^{2}cosh(\frac{j}{kT})} - \frac{N^{2}j(2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N}sinh(\frac{j}{kT})*-sinh(\frac{j}{kT})j*-1}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Tcosh^{2}(\frac{j}{kT})kT^{2}} - \frac{(\frac{-(((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))})){tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N}({tanh(\frac{j}{kT})}^{N}((0)ln(tanh(\frac{j}{kT})) + \frac{(N)(\frac{sech^{2}(\frac{j}{kT})j*-1}{kT^{2}})}{(tanh(\frac{j}{kT}))})) + ((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))})))}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})^{2}})N^{2}j{tanh(\frac{j}{kT})}^{N}(2cosh(\frac{j}{kT}))^{N}sech^{2}(\frac{j}{kT})}{Ttanh(\frac{j}{kT})} - \frac{N^{2}j*-{tanh(\frac{j}{kT})}^{N}(2cosh(\frac{j}{kT}))^{N}sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})T^{2}tanh(\frac{j}{kT})} - \frac{N^{2}j({tanh(\frac{j}{kT})}^{N}((0)ln(tanh(\frac{j}{kT})) + \frac{(N)(\frac{sech^{2}(\frac{j}{kT})j*-1}{kT^{2}})}{(tanh(\frac{j}{kT}))}))(2cosh(\frac{j}{kT}))^{N}sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Ttanh(\frac{j}{kT})} - \frac{N^{2}j{tanh(\frac{j}{kT})}^{N}((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))}))sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Ttanh(\frac{j}{kT})} - \frac{N^{2}j{tanh(\frac{j}{kT})}^{N}(2cosh(\frac{j}{kT}))^{N}*-sech^{2}(\frac{j}{kT})j*-sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Ttanh^{2}(\frac{j}{kT})kT^{2}} - \frac{N^{2}j{tanh(\frac{j}{kT})}^{N}(2cosh(\frac{j}{kT}))^{N}*-2sech(\frac{j}{kT})sech(\frac{j}{kT})tanh(\frac{j}{kT})j*-1}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Ttanh(\frac{j}{kT})kT^{2}} - \frac{(\frac{-(((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))})){tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N}({tanh(\frac{j}{kT})}^{N}((0)ln(tanh(\frac{j}{kT})) + \frac{(N)(\frac{sech^{2}(\frac{j}{kT})j*-1}{kT^{2}})}{(tanh(\frac{j}{kT}))})) + ((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))})))}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})^{2}})N^{2}j(2cosh(\frac{j}{kT}))^{N}sinh(\frac{j}{kT})}{Tcosh(\frac{j}{kT})} - \frac{N^{2}j*-(2cosh(\frac{j}{kT}))^{N}sinh(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})T^{2}cosh(\frac{j}{kT})} - \frac{N^{2}j((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))}))sinh(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Tcosh(\frac{j}{kT})} - \frac{N^{2}j(2cosh(\frac{j}{kT}))^{N}cosh(\frac{j}{kT})j*-1}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})TkT^{2}cosh(\frac{j}{kT})} - \frac{N^{2}j(2cosh(\frac{j}{kT}))^{N}sinh(\frac{j}{kT})*-sinh(\frac{j}{kT})j*-1}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})Tcosh^{2}(\frac{j}{kT})kT^{2}}\\=&\frac{-N^{3}j^{2}(2cosh(\frac{j}{kT}))^{(2N)}{tanh(\frac{j}{kT})}^{(2N)}sinh(\frac{j}{kT})sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})^{2}kT^{3}cosh(\frac{j}{kT})tanh(\frac{j}{kT})} + \frac{N^{3}j^{2}{tanh(\frac{j}{kT})}^{N}(2cosh(\frac{j}{kT}))^{N}sinh(\frac{j}{kT})sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})kT^{3}cosh(\frac{j}{kT})tanh(\frac{j}{kT})} - \frac{N^{3}j^{2}{tanh(\frac{j}{kT})}^{(2N)}(2cosh(\frac{j}{kT}))^{(2N)}sinh(\frac{j}{kT})sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})^{2}kT^{3}cosh(\frac{j}{kT})tanh(\frac{j}{kT})} - \frac{2N^{3}j^{2}(2cosh(\frac{j}{kT}))^{(2N)}{tanh(\frac{j}{kT})}^{N}sinh(\frac{j}{kT})sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})^{2}kT^{3}cosh(\frac{j}{kT})tanh(\frac{j}{kT})} + \frac{N^{3}j^{2}(2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N}sinh(\frac{j}{kT})sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})kT^{3}cosh(\frac{j}{kT})tanh(\frac{j}{kT})} - \frac{N^{2}j^{2}(2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N}sinh^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})kT^{3}cosh^{2}(\frac{j}{kT})} - \frac{N^{2}j^{2}{tanh(\frac{j}{kT})}^{N}(2cosh(\frac{j}{kT}))^{N}sech^{4}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})kT^{3}tanh^{2}(\frac{j}{kT})} - \frac{N^{3}j^{2}(2cosh(\frac{j}{kT}))^{(2N)}{tanh(\frac{j}{kT})}^{N}sinh^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})^{2}kT^{3}cosh^{2}(\frac{j}{kT})} - \frac{N^{3}j^{2}(2cosh(\frac{j}{kT}))^{(2N)}{tanh(\frac{j}{kT})}^{(2N)}sech^{4}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})^{2}kT^{3}tanh^{2}(\frac{j}{kT})} + \frac{N^{3}j^{2}(2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N}sinh^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})kT^{3}cosh^{2}(\frac{j}{kT})} + \frac{N^{3}j^{2}{tanh(\frac{j}{kT})}^{N}(2cosh(\frac{j}{kT}))^{N}sech^{4}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})kT^{3}tanh^{2}(\frac{j}{kT})} - \frac{N^{3}j^{2}{tanh(\frac{j}{kT})}^{(2N)}(2cosh(\frac{j}{kT}))^{(2N)}sinh^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})^{2}kT^{3}cosh^{2}(\frac{j}{kT})} - \frac{2N^{2}j^{2}{tanh(\frac{j}{kT})}^{N}(2cosh(\frac{j}{kT}))^{N}sech^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})kT^{3}} - \frac{N^{3}j^{2}{tanh(\frac{j}{kT})}^{N}(2cosh(\frac{j}{kT}))^{(2N)}sinh^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})^{2}kT^{3}cosh^{2}(\frac{j}{kT})} + \frac{N^{2}j^{2}(2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N}}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})kT^{3}} - \frac{N^{3}j^{2}(2cosh(\frac{j}{kT}))^{(2N)}sinh^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})^{2}kT^{3}cosh^{2}(\frac{j}{kT})} + \frac{N^{3}j^{2}(2cosh(\frac{j}{kT}))^{N}sinh^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})kT^{3}cosh^{2}(\frac{j}{kT})} - \frac{N^{2}j^{2}(2cosh(\frac{j}{kT}))^{N}sinh^{2}(\frac{j}{kT})}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})kT^{3}cosh^{2}(\frac{j}{kT})} + \frac{N^{2}j^{2}(2cosh(\frac{j}{kT}))^{N}}{((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})kT^{3}}\\ \end{split}\end{equation} \]
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