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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ second\ derivative\ of\ function\ -kTln(\frac{1}{2}(1 + cosh(\frac{j}{(kT)}) + {(8 + (2cosh(\frac{j}{(kT)})) - 1)}^{\frac{1}{2}}))\ with\ respect\ to\ T：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = -kTln(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( -kTln(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})\right)}{dT}\\=&-kln(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\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{2sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}}) + 0)}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})}\\=&-kln(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2}) + \frac{jsinh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})T} + \frac{jsinh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}T}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -kln(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2}) + \frac{jsinh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})T} + \frac{jsinh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})(2cosh(\frac{j}{kT}) + 7)^{\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{2sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}}) + 0)}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\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{2sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}}) + 0)}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\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}(2cosh(\frac{j}{kT}) + 7)^{\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}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})TkT^{2}} + \frac{(\frac{-(\frac{\frac{1}{2}sinh(\frac{j}{kT})j*-1}{kT^{2}} + \frac{1}{2}(\frac{\frac{1}{2}(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}}) + 0)}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})^{2}})jsinh(\frac{j}{kT})}{2(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}T} + \frac{(\frac{\frac{-1}{2}(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(2cosh(\frac{j}{kT}) + 7)^{\frac{3}{2}}})jsinh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})T} + \frac{j*-sinh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}T^{2}} + \frac{jcosh(\frac{j}{kT})j*-1}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}TkT^{2}}\\=&\frac{jsinh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}T^{2}} + \frac{j^{2}sinh^{2}(\frac{j}{kT})}{4(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})^{2}kT^{3}} + \frac{j^{2}sinh^{2}(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})^{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}kT^{3}} - \frac{j^{2}cosh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})kT^{3}} + \frac{j^{2}sinh^{2}(\frac{j}{kT})}{4(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})^{2}(2cosh(\frac{j}{kT}) + 7)kT^{3}} + \frac{j^{2}sinh^{2}(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})(2cosh(\frac{j}{kT}) + 7)^{\frac{3}{2}}kT^{3}} - \frac{jsinh(\frac{j}{kT})}{2(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})T^{2}} - \frac{j^{2}cosh(\frac{j}{kT})}{2(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})kT^{3}}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ second\ derivative\ of\ function\ -kTln(\frac{1}{2}(1 + cosh(\frac{j}{(kT)}) + {(8 + (2cosh(\frac{j}{(kT)})) - 1)}^{\frac{1}{2}}))\ with\ respect\ to\ T：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = -kTln(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( -kTln(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})\right)}{dT}\\=&-kln(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\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{2sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}}) + 0)}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})}\\=&-kln(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2}) + \frac{jsinh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})T} + \frac{jsinh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}T}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -kln(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2}) + \frac{jsinh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})T} + \frac{jsinh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})(2cosh(\frac{j}{kT}) + 7)^{\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{2sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}}) + 0)}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\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{2sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}}) + 0)}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\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}(2cosh(\frac{j}{kT}) + 7)^{\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}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})TkT^{2}} + \frac{(\frac{-(\frac{\frac{1}{2}sinh(\frac{j}{kT})j*-1}{kT^{2}} + \frac{1}{2}(\frac{\frac{1}{2}(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}}) + 0)}{(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})^{2}})jsinh(\frac{j}{kT})}{2(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}T} + \frac{(\frac{\frac{-1}{2}(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}} + 0)}{(2cosh(\frac{j}{kT}) + 7)^{\frac{3}{2}}})jsinh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})T} + \frac{j*-sinh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}T^{2}} + \frac{jcosh(\frac{j}{kT})j*-1}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}TkT^{2}}\\=&\frac{jsinh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}T^{2}} + \frac{j^{2}sinh^{2}(\frac{j}{kT})}{4(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})^{2}kT^{3}} + \frac{j^{2}sinh^{2}(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})^{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}kT^{3}} - \frac{j^{2}cosh(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})kT^{3}} + \frac{j^{2}sinh^{2}(\frac{j}{kT})}{4(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})^{2}(2cosh(\frac{j}{kT}) + 7)kT^{3}} + \frac{j^{2}sinh^{2}(\frac{j}{kT})}{2(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})(2cosh(\frac{j}{kT}) + 7)^{\frac{3}{2}}kT^{3}} - \frac{jsinh(\frac{j}{kT})}{2(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})T^{2}} - \frac{j^{2}cosh(\frac{j}{kT})}{2(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}}(\frac{1}{2}cosh(\frac{j}{kT}) + \frac{1}{2}(2cosh(\frac{j}{kT}) + 7)^{\frac{1}{2}} + \frac{1}{2})kT^{3}}\\ \end{split}\end{equation} \]