Derivative function：
Enter an original function (that is, the function to be derived), then set the variable to be derived and the order of the derivative, and click the "Next" button to obtain the derivative function of the corresponding order of the function.
Note that the input function supports mathematical functions and other constants.
Enter an original function (that is, the function to be derived), then set the variable to be derived and the order of the derivative, and click the "Next" button to obtain the derivative function of the corresponding order of the function.
Note that the input function supports mathematical functions and other constants.
Current location：Derivative function > Derivative function calculation history > Answer
There are 1 questions in this calculation: for each question, the 2 derivative of T is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ second\ derivative\ of\ function\ N(kT)ln(({(2cosh(\frac{j}{(kT)}))}^{N})(1 + {(tanh(\frac{j}{(kT)}))}^{N}))\ with\ respect\ to\ T：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = NkTln((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})\\&\color{blue}{The\ first\ derivative\ function：}\\&\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}{The\ second\ derivative\ of\ function:} \\&\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} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ second\ derivative\ of\ function\ N(kT)ln(({(2cosh(\frac{j}{(kT)}))}^{N})(1 + {(tanh(\frac{j}{(kT)}))}^{N}))\ with\ respect\ to\ T：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = NkTln((2cosh(\frac{j}{kT}))^{N}{tanh(\frac{j}{kT})}^{N} + (2cosh(\frac{j}{kT}))^{N})\\&\color{blue}{The\ first\ derivative\ function：}\\&\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}{The\ second\ derivative\ of\ function:} \\&\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} \]