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 4 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ log_{coth(x)}^{tanh(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{coth(x)}^{tanh(x)}\right)}{dx}\\=&(\frac{(\frac{(sech^{2}(x))}{(tanh(x))} - \frac{(-csch^{2}(x))log_{coth(x)}^{tanh(x)}}{(coth(x))})}{(ln(coth(x)))})\\=&\frac{sech^{2}(x)}{ln(coth(x))tanh(x)} + \frac{log_{coth(x)}^{tanh(x)}csch^{2}(x)}{ln(coth(x))coth(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{sech^{2}(x)}{ln(coth(x))tanh(x)} + \frac{log_{coth(x)}^{tanh(x)}csch^{2}(x)}{ln(coth(x))coth(x)}\right)}{dx}\\=&\frac{--csch^{2}(x)sech^{2}(x)}{ln^{2}(coth(x))(coth(x))tanh(x)} + \frac{-sech^{2}(x)sech^{2}(x)}{ln(coth(x))tanh^{2}(x)} + \frac{-2sech(x)sech(x)tanh(x)}{ln(coth(x))tanh(x)} + \frac{(\frac{(\frac{(sech^{2}(x))}{(tanh(x))} - \frac{(-csch^{2}(x))log_{coth(x)}^{tanh(x)}}{(coth(x))})}{(ln(coth(x)))})csch^{2}(x)}{ln(coth(x))coth(x)} + \frac{log_{coth(x)}^{tanh(x)}*--csch^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))(coth(x))coth(x)} + \frac{log_{coth(x)}^{tanh(x)}csch^{2}(x)csch^{2}(x)}{ln(coth(x))coth^{2}(x)} + \frac{log_{coth(x)}^{tanh(x)}*-2csch(x)csch(x)coth(x)}{ln(coth(x))coth(x)}\\=&\frac{2sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh(x)coth(x)} - \frac{sech^{4}(x)}{ln(coth(x))tanh^{2}(x)} - \frac{2sech^{2}(x)}{ln(coth(x))} + \frac{2log_{coth(x)}^{tanh(x)}csch^{4}(x)}{ln^{2}(coth(x))coth^{2}(x)} + \frac{log_{coth(x)}^{tanh(x)}csch^{4}(x)}{ln(coth(x))coth^{2}(x)} - \frac{2log_{coth(x)}^{tanh(x)}csch^{2}(x)}{ln(coth(x))}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh(x)coth(x)} - \frac{sech^{4}(x)}{ln(coth(x))tanh^{2}(x)} - \frac{2sech^{2}(x)}{ln(coth(x))} + \frac{2log_{coth(x)}^{tanh(x)}csch^{4}(x)}{ln^{2}(coth(x))coth^{2}(x)} + \frac{log_{coth(x)}^{tanh(x)}csch^{4}(x)}{ln(coth(x))coth^{2}(x)} - \frac{2log_{coth(x)}^{tanh(x)}csch^{2}(x)}{ln(coth(x))}\right)}{dx}\\=&\frac{2*-2*-csch^{2}(x)sech^{2}(x)csch^{2}(x)}{ln^{3}(coth(x))(coth(x))tanh(x)coth(x)} + \frac{2*-sech^{2}(x)sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh^{2}(x)coth(x)} + \frac{2csch^{2}(x)sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh(x)coth^{2}(x)} + \frac{2*-2sech(x)sech(x)tanh(x)csch^{2}(x)}{ln^{2}(coth(x))tanh(x)coth(x)} + \frac{2sech^{2}(x)*-2csch(x)csch(x)coth(x)}{ln^{2}(coth(x))tanh(x)coth(x)} - \frac{--csch^{2}(x)sech^{4}(x)}{ln^{2}(coth(x))(coth(x))tanh^{2}(x)} - \frac{-2sech^{2}(x)sech^{4}(x)}{ln(coth(x))tanh^{3}(x)} - \frac{-4sech^{3}(x)sech(x)tanh(x)}{ln(coth(x))tanh^{2}(x)} - \frac{2*--csch^{2}(x)sech^{2}(x)}{ln^{2}(coth(x))(coth(x))} - \frac{2*-2sech(x)sech(x)tanh(x)}{ln(coth(x))} + \frac{2(\frac{(\frac{(sech^{2}(x))}{(tanh(x))} - \frac{(-csch^{2}(x))log_{coth(x)}^{tanh(x)}}{(coth(x))})}{(ln(coth(x)))})csch^{4}(x)}{ln^{2}(coth(x))coth^{2}(x)} + \frac{2log_{coth(x)}^{tanh(x)}*-2*-csch^{2}(x)csch^{4}(x)}{ln^{3}(coth(x))(coth(x))coth^{2}(x)} + \frac{2log_{coth(x)}^{tanh(x)}*2csch^{2}(x)csch^{4}(x)}{ln^{2}(coth(x))coth^{3}(x)} + \frac{2log_{coth(x)}^{tanh(x)}*-4csch^{3}(x)csch(x)coth(x)}{ln^{2}(coth(x))coth^{2}(x)} + \frac{(\frac{(\frac{(sech^{2}(x))}{(tanh(x))} - \frac{(-csch^{2}(x))log_{coth(x)}^{tanh(x)}}{(coth(x))})}{(ln(coth(x)))})csch^{4}(x)}{ln(coth(x))coth^{2}(x)} + \frac{log_{coth(x)}^{tanh(x)}*--csch^{2}(x)csch^{4}(x)}{ln^{2}(coth(x))(coth(x))coth^{2}(x)} + \frac{log_{coth(x)}^{tanh(x)}*2csch^{2}(x)csch^{4}(x)}{ln(coth(x))coth^{3}(x)} + \frac{log_{coth(x)}^{tanh(x)}*-4csch^{3}(x)csch(x)coth(x)}{ln(coth(x))coth^{2}(x)} - \frac{2(\frac{(\frac{(sech^{2}(x))}{(tanh(x))} - \frac{(-csch^{2}(x))log_{coth(x)}^{tanh(x)}}{(coth(x))})}{(ln(coth(x)))})csch^{2}(x)}{ln(coth(x))} - \frac{2log_{coth(x)}^{tanh(x)}*--csch^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))(coth(x))} - \frac{2log_{coth(x)}^{tanh(x)}*-2csch(x)csch(x)coth(x)}{ln(coth(x))}\\=&\frac{6sech^{2}(x)csch^{4}(x)}{ln^{3}(coth(x))tanh(x)coth^{2}(x)} - \frac{3sech^{4}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh^{2}(x)coth(x)} + \frac{3sech^{2}(x)csch^{4}(x)}{ln^{2}(coth(x))tanh(x)coth^{2}(x)} - \frac{6sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))coth(x)} - \frac{6sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh(x)} + \frac{2sech^{6}(x)}{ln(coth(x))tanh^{3}(x)} + \frac{4sech^{4}(x)}{ln(coth(x))tanh(x)} + \frac{4tanh(x)sech^{2}(x)}{ln(coth(x))} + \frac{6log_{coth(x)}^{tanh(x)}csch^{6}(x)}{ln^{3}(coth(x))coth^{3}(x)} + \frac{6log_{coth(x)}^{tanh(x)}csch^{6}(x)}{ln^{2}(coth(x))coth^{3}(x)} - \frac{12log_{coth(x)}^{tanh(x)}csch^{4}(x)}{ln^{2}(coth(x))coth(x)} + \frac{2log_{coth(x)}^{tanh(x)}csch^{6}(x)}{ln(coth(x))coth^{3}(x)} - \frac{4log_{coth(x)}^{tanh(x)}csch^{4}(x)}{ln(coth(x))coth(x)} + \frac{4log_{coth(x)}^{tanh(x)}coth(x)csch^{2}(x)}{ln(coth(x))}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{6sech^{2}(x)csch^{4}(x)}{ln^{3}(coth(x))tanh(x)coth^{2}(x)} - \frac{3sech^{4}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh^{2}(x)coth(x)} + \frac{3sech^{2}(x)csch^{4}(x)}{ln^{2}(coth(x))tanh(x)coth^{2}(x)} - \frac{6sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))coth(x)} - \frac{6sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh(x)} + \frac{2sech^{6}(x)}{ln(coth(x))tanh^{3}(x)} + \frac{4sech^{4}(x)}{ln(coth(x))tanh(x)} + \frac{4tanh(x)sech^{2}(x)}{ln(coth(x))} + \frac{6log_{coth(x)}^{tanh(x)}csch^{6}(x)}{ln^{3}(coth(x))coth^{3}(x)} + \frac{6log_{coth(x)}^{tanh(x)}csch^{6}(x)}{ln^{2}(coth(x))coth^{3}(x)} - \frac{12log_{coth(x)}^{tanh(x)}csch^{4}(x)}{ln^{2}(coth(x))coth(x)} + \frac{2log_{coth(x)}^{tanh(x)}csch^{6}(x)}{ln(coth(x))coth^{3}(x)} - \frac{4log_{coth(x)}^{tanh(x)}csch^{4}(x)}{ln(coth(x))coth(x)} + \frac{4log_{coth(x)}^{tanh(x)}coth(x)csch^{2}(x)}{ln(coth(x))}\right)}{dx}\\=&\frac{6*-3*-csch^{2}(x)sech^{2}(x)csch^{4}(x)}{ln^{4}(coth(x))(coth(x))tanh(x)coth^{2}(x)} + \frac{6*-sech^{2}(x)sech^{2}(x)csch^{4}(x)}{ln^{3}(coth(x))tanh^{2}(x)coth^{2}(x)} + \frac{6*2csch^{2}(x)sech^{2}(x)csch^{4}(x)}{ln^{3}(coth(x))tanh(x)coth^{3}(x)} + \frac{6*-2sech(x)sech(x)tanh(x)csch^{4}(x)}{ln^{3}(coth(x))tanh(x)coth^{2}(x)} + \frac{6sech^{2}(x)*-4csch^{3}(x)csch(x)coth(x)}{ln^{3}(coth(x))tanh(x)coth^{2}(x)} - \frac{3*-2*-csch^{2}(x)sech^{4}(x)csch^{2}(x)}{ln^{3}(coth(x))(coth(x))tanh^{2}(x)coth(x)} - \frac{3*-2sech^{2}(x)sech^{4}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh^{3}(x)coth(x)} - \frac{3csch^{2}(x)sech^{4}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh^{2}(x)coth^{2}(x)} - \frac{3*-4sech^{3}(x)sech(x)tanh(x)csch^{2}(x)}{ln^{2}(coth(x))tanh^{2}(x)coth(x)} - \frac{3sech^{4}(x)*-2csch(x)csch(x)coth(x)}{ln^{2}(coth(x))tanh^{2}(x)coth(x)} + \frac{3*-2*-csch^{2}(x)sech^{2}(x)csch^{4}(x)}{ln^{3}(coth(x))(coth(x))tanh(x)coth^{2}(x)} + \frac{3*-sech^{2}(x)sech^{2}(x)csch^{4}(x)}{ln^{2}(coth(x))tanh^{2}(x)coth^{2}(x)} + \frac{3*2csch^{2}(x)sech^{2}(x)csch^{4}(x)}{ln^{2}(coth(x))tanh(x)coth^{3}(x)} + \frac{3*-2sech(x)sech(x)tanh(x)csch^{4}(x)}{ln^{2}(coth(x))tanh(x)coth^{2}(x)} + \frac{3sech^{2}(x)*-4csch^{3}(x)csch(x)coth(x)}{ln^{2}(coth(x))tanh(x)coth^{2}(x)} - \frac{6*-2*-csch^{2}(x)sech^{2}(x)csch^{2}(x)}{ln^{3}(coth(x))(coth(x))coth(x)} - \frac{6csch^{2}(x)sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))coth^{2}(x)} - \frac{6*-2sech(x)sech(x)tanh(x)csch^{2}(x)}{ln^{2}(coth(x))coth(x)} - \frac{6sech^{2}(x)*-2csch(x)csch(x)coth(x)}{ln^{2}(coth(x))coth(x)} - \frac{6*-2*-csch^{2}(x)sech^{2}(x)csch^{2}(x)}{ln^{3}(coth(x))(coth(x))tanh(x)} - \frac{6*-sech^{2}(x)sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh^{2}(x)} - \frac{6*-2sech(x)sech(x)tanh(x)csch^{2}(x)}{ln^{2}(coth(x))tanh(x)} - \frac{6sech^{2}(x)*-2csch(x)csch(x)coth(x)}{ln^{2}(coth(x))tanh(x)} + \frac{2*--csch^{2}(x)sech^{6}(x)}{ln^{2}(coth(x))(coth(x))tanh^{3}(x)} + \frac{2*-3sech^{2}(x)sech^{6}(x)}{ln(coth(x))tanh^{4}(x)} + \frac{2*-6sech^{5}(x)sech(x)tanh(x)}{ln(coth(x))tanh^{3}(x)} + \frac{4*--csch^{2}(x)sech^{4}(x)}{ln^{2}(coth(x))(coth(x))tanh(x)} + \frac{4*-sech^{2}(x)sech^{4}(x)}{ln(coth(x))tanh^{2}(x)} + \frac{4*-4sech^{3}(x)sech(x)tanh(x)}{ln(coth(x))tanh(x)} + \frac{4*--csch^{2}(x)tanh(x)sech^{2}(x)}{ln^{2}(coth(x))(coth(x))} + \frac{4sech^{2}(x)sech^{2}(x)}{ln(coth(x))} + \frac{4tanh(x)*-2sech(x)sech(x)tanh(x)}{ln(coth(x))} + \frac{6(\frac{(\frac{(sech^{2}(x))}{(tanh(x))} - \frac{(-csch^{2}(x))log_{coth(x)}^{tanh(x)}}{(coth(x))})}{(ln(coth(x)))})csch^{6}(x)}{ln^{3}(coth(x))coth^{3}(x)} + \frac{6log_{coth(x)}^{tanh(x)}*-3*-csch^{2}(x)csch^{6}(x)}{ln^{4}(coth(x))(coth(x))coth^{3}(x)} + \frac{6log_{coth(x)}^{tanh(x)}*3csch^{2}(x)csch^{6}(x)}{ln^{3}(coth(x))coth^{4}(x)} + \frac{6log_{coth(x)}^{tanh(x)}*-6csch^{5}(x)csch(x)coth(x)}{ln^{3}(coth(x))coth^{3}(x)} + \frac{6(\frac{(\frac{(sech^{2}(x))}{(tanh(x))} - \frac{(-csch^{2}(x))log_{coth(x)}^{tanh(x)}}{(coth(x))})}{(ln(coth(x)))})csch^{6}(x)}{ln^{2}(coth(x))coth^{3}(x)} + \frac{6log_{coth(x)}^{tanh(x)}*-2*-csch^{2}(x)csch^{6}(x)}{ln^{3}(coth(x))(coth(x))coth^{3}(x)} + \frac{6log_{coth(x)}^{tanh(x)}*3csch^{2}(x)csch^{6}(x)}{ln^{2}(coth(x))coth^{4}(x)} + \frac{6log_{coth(x)}^{tanh(x)}*-6csch^{5}(x)csch(x)coth(x)}{ln^{2}(coth(x))coth^{3}(x)} - \frac{12(\frac{(\frac{(sech^{2}(x))}{(tanh(x))} - \frac{(-csch^{2}(x))log_{coth(x)}^{tanh(x)}}{(coth(x))})}{(ln(coth(x)))})csch^{4}(x)}{ln^{2}(coth(x))coth(x)} - \frac{12log_{coth(x)}^{tanh(x)}*-2*-csch^{2}(x)csch^{4}(x)}{ln^{3}(coth(x))(coth(x))coth(x)} - \frac{12log_{coth(x)}^{tanh(x)}csch^{2}(x)csch^{4}(x)}{ln^{2}(coth(x))coth^{2}(x)} - \frac{12log_{coth(x)}^{tanh(x)}*-4csch^{3}(x)csch(x)coth(x)}{ln^{2}(coth(x))coth(x)} + \frac{2(\frac{(\frac{(sech^{2}(x))}{(tanh(x))} - \frac{(-csch^{2}(x))log_{coth(x)}^{tanh(x)}}{(coth(x))})}{(ln(coth(x)))})csch^{6}(x)}{ln(coth(x))coth^{3}(x)} + \frac{2log_{coth(x)}^{tanh(x)}*--csch^{2}(x)csch^{6}(x)}{ln^{2}(coth(x))(coth(x))coth^{3}(x)} + \frac{2log_{coth(x)}^{tanh(x)}*3csch^{2}(x)csch^{6}(x)}{ln(coth(x))coth^{4}(x)} + \frac{2log_{coth(x)}^{tanh(x)}*-6csch^{5}(x)csch(x)coth(x)}{ln(coth(x))coth^{3}(x)} - \frac{4(\frac{(\frac{(sech^{2}(x))}{(tanh(x))} - \frac{(-csch^{2}(x))log_{coth(x)}^{tanh(x)}}{(coth(x))})}{(ln(coth(x)))})csch^{4}(x)}{ln(coth(x))coth(x)} - \frac{4log_{coth(x)}^{tanh(x)}*--csch^{2}(x)csch^{4}(x)}{ln^{2}(coth(x))(coth(x))coth(x)} - \frac{4log_{coth(x)}^{tanh(x)}csch^{2}(x)csch^{4}(x)}{ln(coth(x))coth^{2}(x)} - \frac{4log_{coth(x)}^{tanh(x)}*-4csch^{3}(x)csch(x)coth(x)}{ln(coth(x))coth(x)} + \frac{4(\frac{(\frac{(sech^{2}(x))}{(tanh(x))} - \frac{(-csch^{2}(x))log_{coth(x)}^{tanh(x)}}{(coth(x))})}{(ln(coth(x)))})coth(x)csch^{2}(x)}{ln(coth(x))} + \frac{4log_{coth(x)}^{tanh(x)}*--csch^{2}(x)coth(x)csch^{2}(x)}{ln^{2}(coth(x))(coth(x))} + \frac{4log_{coth(x)}^{tanh(x)}*-csch^{2}(x)csch^{2}(x)}{ln(coth(x))} + \frac{4log_{coth(x)}^{tanh(x)}coth(x)*-2csch(x)csch(x)coth(x)}{ln(coth(x))}\\=&\frac{24sech^{2}(x)csch^{6}(x)}{ln^{4}(coth(x))tanh(x)coth^{3}(x)} - \frac{12sech^{4}(x)csch^{4}(x)}{ln^{3}(coth(x))tanh^{2}(x)coth^{2}(x)} + \frac{24sech^{2}(x)csch^{6}(x)}{ln^{3}(coth(x))tanh(x)coth^{3}(x)} - \frac{24sech^{2}(x)csch^{4}(x)}{ln^{3}(coth(x))coth^{2}(x)} - \frac{48sech^{2}(x)csch^{4}(x)}{ln^{3}(coth(x))tanh(x)coth(x)} + \frac{8sech^{6}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh^{3}(x)coth(x)} - \frac{6sech^{4}(x)csch^{4}(x)}{ln^{2}(coth(x))tanh^{2}(x)coth^{2}(x)} + \frac{16sech^{4}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh(x)coth(x)} + \frac{12sech^{4}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh^{2}(x)} + \frac{8sech^{2}(x)csch^{6}(x)}{ln^{2}(coth(x))tanh(x)coth^{3}(x)} - \frac{12sech^{2}(x)csch^{4}(x)}{ln^{2}(coth(x))coth^{2}(x)} - \frac{16sech^{2}(x)csch^{4}(x)}{ln^{2}(coth(x))tanh(x)coth(x)} + \frac{16tanh(x)sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))coth(x)} + \frac{24sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))} + \frac{16coth(x)sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh(x)} - \frac{6sech^{8}(x)}{ln(coth(x))tanh^{4}(x)} - \frac{16sech^{6}(x)}{ln(coth(x))tanh^{2}(x)} - \frac{12sech^{4}(x)}{ln(coth(x))} - \frac{8tanh^{2}(x)sech^{2}(x)}{ln(coth(x))} + \frac{24log_{coth(x)}^{tanh(x)}csch^{8}(x)}{ln^{4}(coth(x))coth^{4}(x)} + \frac{36log_{coth(x)}^{tanh(x)}csch^{8}(x)}{ln^{3}(coth(x))coth^{4}(x)} - \frac{72log_{coth(x)}^{tanh(x)}csch^{6}(x)}{ln^{3}(coth(x))coth^{2}(x)} + \frac{22log_{coth(x)}^{tanh(x)}csch^{8}(x)}{ln^{2}(coth(x))coth^{4}(x)} - \frac{56log_{coth(x)}^{tanh(x)}csch^{6}(x)}{ln^{2}(coth(x))coth^{2}(x)} + \frac{56log_{coth(x)}^{tanh(x)}csch^{4}(x)}{ln^{2}(coth(x))} + \frac{6log_{coth(x)}^{tanh(x)}csch^{8}(x)}{ln(coth(x))coth^{4}(x)} - \frac{16log_{coth(x)}^{tanh(x)}csch^{6}(x)}{ln(coth(x))coth^{2}(x)} + \frac{12log_{coth(x)}^{tanh(x)}csch^{4}(x)}{ln(coth(x))} - \frac{8log_{coth(x)}^{tanh(x)}coth^{2}(x)csch^{2}(x)}{ln(coth(x))}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ log_{coth(x)}^{tanh(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{coth(x)}^{tanh(x)}\right)}{dx}\\=&(\frac{(\frac{(sech^{2}(x))}{(tanh(x))} - \frac{(-csch^{2}(x))log_{coth(x)}^{tanh(x)}}{(coth(x))})}{(ln(coth(x)))})\\=&\frac{sech^{2}(x)}{ln(coth(x))tanh(x)} + \frac{log_{coth(x)}^{tanh(x)}csch^{2}(x)}{ln(coth(x))coth(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{sech^{2}(x)}{ln(coth(x))tanh(x)} + \frac{log_{coth(x)}^{tanh(x)}csch^{2}(x)}{ln(coth(x))coth(x)}\right)}{dx}\\=&\frac{--csch^{2}(x)sech^{2}(x)}{ln^{2}(coth(x))(coth(x))tanh(x)} + \frac{-sech^{2}(x)sech^{2}(x)}{ln(coth(x))tanh^{2}(x)} + \frac{-2sech(x)sech(x)tanh(x)}{ln(coth(x))tanh(x)} + \frac{(\frac{(\frac{(sech^{2}(x))}{(tanh(x))} - \frac{(-csch^{2}(x))log_{coth(x)}^{tanh(x)}}{(coth(x))})}{(ln(coth(x)))})csch^{2}(x)}{ln(coth(x))coth(x)} + \frac{log_{coth(x)}^{tanh(x)}*--csch^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))(coth(x))coth(x)} + \frac{log_{coth(x)}^{tanh(x)}csch^{2}(x)csch^{2}(x)}{ln(coth(x))coth^{2}(x)} + \frac{log_{coth(x)}^{tanh(x)}*-2csch(x)csch(x)coth(x)}{ln(coth(x))coth(x)}\\=&\frac{2sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh(x)coth(x)} - \frac{sech^{4}(x)}{ln(coth(x))tanh^{2}(x)} - \frac{2sech^{2}(x)}{ln(coth(x))} + \frac{2log_{coth(x)}^{tanh(x)}csch^{4}(x)}{ln^{2}(coth(x))coth^{2}(x)} + \frac{log_{coth(x)}^{tanh(x)}csch^{4}(x)}{ln(coth(x))coth^{2}(x)} - \frac{2log_{coth(x)}^{tanh(x)}csch^{2}(x)}{ln(coth(x))}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh(x)coth(x)} - \frac{sech^{4}(x)}{ln(coth(x))tanh^{2}(x)} - \frac{2sech^{2}(x)}{ln(coth(x))} + \frac{2log_{coth(x)}^{tanh(x)}csch^{4}(x)}{ln^{2}(coth(x))coth^{2}(x)} + \frac{log_{coth(x)}^{tanh(x)}csch^{4}(x)}{ln(coth(x))coth^{2}(x)} - \frac{2log_{coth(x)}^{tanh(x)}csch^{2}(x)}{ln(coth(x))}\right)}{dx}\\=&\frac{2*-2*-csch^{2}(x)sech^{2}(x)csch^{2}(x)}{ln^{3}(coth(x))(coth(x))tanh(x)coth(x)} + \frac{2*-sech^{2}(x)sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh^{2}(x)coth(x)} + \frac{2csch^{2}(x)sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh(x)coth^{2}(x)} + \frac{2*-2sech(x)sech(x)tanh(x)csch^{2}(x)}{ln^{2}(coth(x))tanh(x)coth(x)} + \frac{2sech^{2}(x)*-2csch(x)csch(x)coth(x)}{ln^{2}(coth(x))tanh(x)coth(x)} - \frac{--csch^{2}(x)sech^{4}(x)}{ln^{2}(coth(x))(coth(x))tanh^{2}(x)} - \frac{-2sech^{2}(x)sech^{4}(x)}{ln(coth(x))tanh^{3}(x)} - \frac{-4sech^{3}(x)sech(x)tanh(x)}{ln(coth(x))tanh^{2}(x)} - \frac{2*--csch^{2}(x)sech^{2}(x)}{ln^{2}(coth(x))(coth(x))} - \frac{2*-2sech(x)sech(x)tanh(x)}{ln(coth(x))} + \frac{2(\frac{(\frac{(sech^{2}(x))}{(tanh(x))} - \frac{(-csch^{2}(x))log_{coth(x)}^{tanh(x)}}{(coth(x))})}{(ln(coth(x)))})csch^{4}(x)}{ln^{2}(coth(x))coth^{2}(x)} + \frac{2log_{coth(x)}^{tanh(x)}*-2*-csch^{2}(x)csch^{4}(x)}{ln^{3}(coth(x))(coth(x))coth^{2}(x)} + \frac{2log_{coth(x)}^{tanh(x)}*2csch^{2}(x)csch^{4}(x)}{ln^{2}(coth(x))coth^{3}(x)} + \frac{2log_{coth(x)}^{tanh(x)}*-4csch^{3}(x)csch(x)coth(x)}{ln^{2}(coth(x))coth^{2}(x)} + \frac{(\frac{(\frac{(sech^{2}(x))}{(tanh(x))} - \frac{(-csch^{2}(x))log_{coth(x)}^{tanh(x)}}{(coth(x))})}{(ln(coth(x)))})csch^{4}(x)}{ln(coth(x))coth^{2}(x)} + \frac{log_{coth(x)}^{tanh(x)}*--csch^{2}(x)csch^{4}(x)}{ln^{2}(coth(x))(coth(x))coth^{2}(x)} + \frac{log_{coth(x)}^{tanh(x)}*2csch^{2}(x)csch^{4}(x)}{ln(coth(x))coth^{3}(x)} + \frac{log_{coth(x)}^{tanh(x)}*-4csch^{3}(x)csch(x)coth(x)}{ln(coth(x))coth^{2}(x)} - \frac{2(\frac{(\frac{(sech^{2}(x))}{(tanh(x))} - \frac{(-csch^{2}(x))log_{coth(x)}^{tanh(x)}}{(coth(x))})}{(ln(coth(x)))})csch^{2}(x)}{ln(coth(x))} - \frac{2log_{coth(x)}^{tanh(x)}*--csch^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))(coth(x))} - \frac{2log_{coth(x)}^{tanh(x)}*-2csch(x)csch(x)coth(x)}{ln(coth(x))}\\=&\frac{6sech^{2}(x)csch^{4}(x)}{ln^{3}(coth(x))tanh(x)coth^{2}(x)} - \frac{3sech^{4}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh^{2}(x)coth(x)} + \frac{3sech^{2}(x)csch^{4}(x)}{ln^{2}(coth(x))tanh(x)coth^{2}(x)} - \frac{6sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))coth(x)} - \frac{6sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh(x)} + \frac{2sech^{6}(x)}{ln(coth(x))tanh^{3}(x)} + \frac{4sech^{4}(x)}{ln(coth(x))tanh(x)} + \frac{4tanh(x)sech^{2}(x)}{ln(coth(x))} + \frac{6log_{coth(x)}^{tanh(x)}csch^{6}(x)}{ln^{3}(coth(x))coth^{3}(x)} + \frac{6log_{coth(x)}^{tanh(x)}csch^{6}(x)}{ln^{2}(coth(x))coth^{3}(x)} - \frac{12log_{coth(x)}^{tanh(x)}csch^{4}(x)}{ln^{2}(coth(x))coth(x)} + \frac{2log_{coth(x)}^{tanh(x)}csch^{6}(x)}{ln(coth(x))coth^{3}(x)} - \frac{4log_{coth(x)}^{tanh(x)}csch^{4}(x)}{ln(coth(x))coth(x)} + \frac{4log_{coth(x)}^{tanh(x)}coth(x)csch^{2}(x)}{ln(coth(x))}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{6sech^{2}(x)csch^{4}(x)}{ln^{3}(coth(x))tanh(x)coth^{2}(x)} - \frac{3sech^{4}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh^{2}(x)coth(x)} + \frac{3sech^{2}(x)csch^{4}(x)}{ln^{2}(coth(x))tanh(x)coth^{2}(x)} - \frac{6sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))coth(x)} - \frac{6sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh(x)} + \frac{2sech^{6}(x)}{ln(coth(x))tanh^{3}(x)} + \frac{4sech^{4}(x)}{ln(coth(x))tanh(x)} + \frac{4tanh(x)sech^{2}(x)}{ln(coth(x))} + \frac{6log_{coth(x)}^{tanh(x)}csch^{6}(x)}{ln^{3}(coth(x))coth^{3}(x)} + \frac{6log_{coth(x)}^{tanh(x)}csch^{6}(x)}{ln^{2}(coth(x))coth^{3}(x)} - \frac{12log_{coth(x)}^{tanh(x)}csch^{4}(x)}{ln^{2}(coth(x))coth(x)} + \frac{2log_{coth(x)}^{tanh(x)}csch^{6}(x)}{ln(coth(x))coth^{3}(x)} - \frac{4log_{coth(x)}^{tanh(x)}csch^{4}(x)}{ln(coth(x))coth(x)} + \frac{4log_{coth(x)}^{tanh(x)}coth(x)csch^{2}(x)}{ln(coth(x))}\right)}{dx}\\=&\frac{6*-3*-csch^{2}(x)sech^{2}(x)csch^{4}(x)}{ln^{4}(coth(x))(coth(x))tanh(x)coth^{2}(x)} + \frac{6*-sech^{2}(x)sech^{2}(x)csch^{4}(x)}{ln^{3}(coth(x))tanh^{2}(x)coth^{2}(x)} + \frac{6*2csch^{2}(x)sech^{2}(x)csch^{4}(x)}{ln^{3}(coth(x))tanh(x)coth^{3}(x)} + \frac{6*-2sech(x)sech(x)tanh(x)csch^{4}(x)}{ln^{3}(coth(x))tanh(x)coth^{2}(x)} + \frac{6sech^{2}(x)*-4csch^{3}(x)csch(x)coth(x)}{ln^{3}(coth(x))tanh(x)coth^{2}(x)} - \frac{3*-2*-csch^{2}(x)sech^{4}(x)csch^{2}(x)}{ln^{3}(coth(x))(coth(x))tanh^{2}(x)coth(x)} - \frac{3*-2sech^{2}(x)sech^{4}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh^{3}(x)coth(x)} - \frac{3csch^{2}(x)sech^{4}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh^{2}(x)coth^{2}(x)} - \frac{3*-4sech^{3}(x)sech(x)tanh(x)csch^{2}(x)}{ln^{2}(coth(x))tanh^{2}(x)coth(x)} - \frac{3sech^{4}(x)*-2csch(x)csch(x)coth(x)}{ln^{2}(coth(x))tanh^{2}(x)coth(x)} + \frac{3*-2*-csch^{2}(x)sech^{2}(x)csch^{4}(x)}{ln^{3}(coth(x))(coth(x))tanh(x)coth^{2}(x)} + \frac{3*-sech^{2}(x)sech^{2}(x)csch^{4}(x)}{ln^{2}(coth(x))tanh^{2}(x)coth^{2}(x)} + \frac{3*2csch^{2}(x)sech^{2}(x)csch^{4}(x)}{ln^{2}(coth(x))tanh(x)coth^{3}(x)} + \frac{3*-2sech(x)sech(x)tanh(x)csch^{4}(x)}{ln^{2}(coth(x))tanh(x)coth^{2}(x)} + \frac{3sech^{2}(x)*-4csch^{3}(x)csch(x)coth(x)}{ln^{2}(coth(x))tanh(x)coth^{2}(x)} - \frac{6*-2*-csch^{2}(x)sech^{2}(x)csch^{2}(x)}{ln^{3}(coth(x))(coth(x))coth(x)} - \frac{6csch^{2}(x)sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))coth^{2}(x)} - \frac{6*-2sech(x)sech(x)tanh(x)csch^{2}(x)}{ln^{2}(coth(x))coth(x)} - \frac{6sech^{2}(x)*-2csch(x)csch(x)coth(x)}{ln^{2}(coth(x))coth(x)} - \frac{6*-2*-csch^{2}(x)sech^{2}(x)csch^{2}(x)}{ln^{3}(coth(x))(coth(x))tanh(x)} - \frac{6*-sech^{2}(x)sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh^{2}(x)} - \frac{6*-2sech(x)sech(x)tanh(x)csch^{2}(x)}{ln^{2}(coth(x))tanh(x)} - \frac{6sech^{2}(x)*-2csch(x)csch(x)coth(x)}{ln^{2}(coth(x))tanh(x)} + \frac{2*--csch^{2}(x)sech^{6}(x)}{ln^{2}(coth(x))(coth(x))tanh^{3}(x)} + \frac{2*-3sech^{2}(x)sech^{6}(x)}{ln(coth(x))tanh^{4}(x)} + \frac{2*-6sech^{5}(x)sech(x)tanh(x)}{ln(coth(x))tanh^{3}(x)} + \frac{4*--csch^{2}(x)sech^{4}(x)}{ln^{2}(coth(x))(coth(x))tanh(x)} + \frac{4*-sech^{2}(x)sech^{4}(x)}{ln(coth(x))tanh^{2}(x)} + \frac{4*-4sech^{3}(x)sech(x)tanh(x)}{ln(coth(x))tanh(x)} + \frac{4*--csch^{2}(x)tanh(x)sech^{2}(x)}{ln^{2}(coth(x))(coth(x))} + \frac{4sech^{2}(x)sech^{2}(x)}{ln(coth(x))} + \frac{4tanh(x)*-2sech(x)sech(x)tanh(x)}{ln(coth(x))} + \frac{6(\frac{(\frac{(sech^{2}(x))}{(tanh(x))} - \frac{(-csch^{2}(x))log_{coth(x)}^{tanh(x)}}{(coth(x))})}{(ln(coth(x)))})csch^{6}(x)}{ln^{3}(coth(x))coth^{3}(x)} + \frac{6log_{coth(x)}^{tanh(x)}*-3*-csch^{2}(x)csch^{6}(x)}{ln^{4}(coth(x))(coth(x))coth^{3}(x)} + \frac{6log_{coth(x)}^{tanh(x)}*3csch^{2}(x)csch^{6}(x)}{ln^{3}(coth(x))coth^{4}(x)} + \frac{6log_{coth(x)}^{tanh(x)}*-6csch^{5}(x)csch(x)coth(x)}{ln^{3}(coth(x))coth^{3}(x)} + \frac{6(\frac{(\frac{(sech^{2}(x))}{(tanh(x))} - \frac{(-csch^{2}(x))log_{coth(x)}^{tanh(x)}}{(coth(x))})}{(ln(coth(x)))})csch^{6}(x)}{ln^{2}(coth(x))coth^{3}(x)} + \frac{6log_{coth(x)}^{tanh(x)}*-2*-csch^{2}(x)csch^{6}(x)}{ln^{3}(coth(x))(coth(x))coth^{3}(x)} + \frac{6log_{coth(x)}^{tanh(x)}*3csch^{2}(x)csch^{6}(x)}{ln^{2}(coth(x))coth^{4}(x)} + \frac{6log_{coth(x)}^{tanh(x)}*-6csch^{5}(x)csch(x)coth(x)}{ln^{2}(coth(x))coth^{3}(x)} - \frac{12(\frac{(\frac{(sech^{2}(x))}{(tanh(x))} - \frac{(-csch^{2}(x))log_{coth(x)}^{tanh(x)}}{(coth(x))})}{(ln(coth(x)))})csch^{4}(x)}{ln^{2}(coth(x))coth(x)} - \frac{12log_{coth(x)}^{tanh(x)}*-2*-csch^{2}(x)csch^{4}(x)}{ln^{3}(coth(x))(coth(x))coth(x)} - \frac{12log_{coth(x)}^{tanh(x)}csch^{2}(x)csch^{4}(x)}{ln^{2}(coth(x))coth^{2}(x)} - \frac{12log_{coth(x)}^{tanh(x)}*-4csch^{3}(x)csch(x)coth(x)}{ln^{2}(coth(x))coth(x)} + \frac{2(\frac{(\frac{(sech^{2}(x))}{(tanh(x))} - \frac{(-csch^{2}(x))log_{coth(x)}^{tanh(x)}}{(coth(x))})}{(ln(coth(x)))})csch^{6}(x)}{ln(coth(x))coth^{3}(x)} + \frac{2log_{coth(x)}^{tanh(x)}*--csch^{2}(x)csch^{6}(x)}{ln^{2}(coth(x))(coth(x))coth^{3}(x)} + \frac{2log_{coth(x)}^{tanh(x)}*3csch^{2}(x)csch^{6}(x)}{ln(coth(x))coth^{4}(x)} + \frac{2log_{coth(x)}^{tanh(x)}*-6csch^{5}(x)csch(x)coth(x)}{ln(coth(x))coth^{3}(x)} - \frac{4(\frac{(\frac{(sech^{2}(x))}{(tanh(x))} - \frac{(-csch^{2}(x))log_{coth(x)}^{tanh(x)}}{(coth(x))})}{(ln(coth(x)))})csch^{4}(x)}{ln(coth(x))coth(x)} - \frac{4log_{coth(x)}^{tanh(x)}*--csch^{2}(x)csch^{4}(x)}{ln^{2}(coth(x))(coth(x))coth(x)} - \frac{4log_{coth(x)}^{tanh(x)}csch^{2}(x)csch^{4}(x)}{ln(coth(x))coth^{2}(x)} - \frac{4log_{coth(x)}^{tanh(x)}*-4csch^{3}(x)csch(x)coth(x)}{ln(coth(x))coth(x)} + \frac{4(\frac{(\frac{(sech^{2}(x))}{(tanh(x))} - \frac{(-csch^{2}(x))log_{coth(x)}^{tanh(x)}}{(coth(x))})}{(ln(coth(x)))})coth(x)csch^{2}(x)}{ln(coth(x))} + \frac{4log_{coth(x)}^{tanh(x)}*--csch^{2}(x)coth(x)csch^{2}(x)}{ln^{2}(coth(x))(coth(x))} + \frac{4log_{coth(x)}^{tanh(x)}*-csch^{2}(x)csch^{2}(x)}{ln(coth(x))} + \frac{4log_{coth(x)}^{tanh(x)}coth(x)*-2csch(x)csch(x)coth(x)}{ln(coth(x))}\\=&\frac{24sech^{2}(x)csch^{6}(x)}{ln^{4}(coth(x))tanh(x)coth^{3}(x)} - \frac{12sech^{4}(x)csch^{4}(x)}{ln^{3}(coth(x))tanh^{2}(x)coth^{2}(x)} + \frac{24sech^{2}(x)csch^{6}(x)}{ln^{3}(coth(x))tanh(x)coth^{3}(x)} - \frac{24sech^{2}(x)csch^{4}(x)}{ln^{3}(coth(x))coth^{2}(x)} - \frac{48sech^{2}(x)csch^{4}(x)}{ln^{3}(coth(x))tanh(x)coth(x)} + \frac{8sech^{6}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh^{3}(x)coth(x)} - \frac{6sech^{4}(x)csch^{4}(x)}{ln^{2}(coth(x))tanh^{2}(x)coth^{2}(x)} + \frac{16sech^{4}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh(x)coth(x)} + \frac{12sech^{4}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh^{2}(x)} + \frac{8sech^{2}(x)csch^{6}(x)}{ln^{2}(coth(x))tanh(x)coth^{3}(x)} - \frac{12sech^{2}(x)csch^{4}(x)}{ln^{2}(coth(x))coth^{2}(x)} - \frac{16sech^{2}(x)csch^{4}(x)}{ln^{2}(coth(x))tanh(x)coth(x)} + \frac{16tanh(x)sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))coth(x)} + \frac{24sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))} + \frac{16coth(x)sech^{2}(x)csch^{2}(x)}{ln^{2}(coth(x))tanh(x)} - \frac{6sech^{8}(x)}{ln(coth(x))tanh^{4}(x)} - \frac{16sech^{6}(x)}{ln(coth(x))tanh^{2}(x)} - \frac{12sech^{4}(x)}{ln(coth(x))} - \frac{8tanh^{2}(x)sech^{2}(x)}{ln(coth(x))} + \frac{24log_{coth(x)}^{tanh(x)}csch^{8}(x)}{ln^{4}(coth(x))coth^{4}(x)} + \frac{36log_{coth(x)}^{tanh(x)}csch^{8}(x)}{ln^{3}(coth(x))coth^{4}(x)} - \frac{72log_{coth(x)}^{tanh(x)}csch^{6}(x)}{ln^{3}(coth(x))coth^{2}(x)} + \frac{22log_{coth(x)}^{tanh(x)}csch^{8}(x)}{ln^{2}(coth(x))coth^{4}(x)} - \frac{56log_{coth(x)}^{tanh(x)}csch^{6}(x)}{ln^{2}(coth(x))coth^{2}(x)} + \frac{56log_{coth(x)}^{tanh(x)}csch^{4}(x)}{ln^{2}(coth(x))} + \frac{6log_{coth(x)}^{tanh(x)}csch^{8}(x)}{ln(coth(x))coth^{4}(x)} - \frac{16log_{coth(x)}^{tanh(x)}csch^{6}(x)}{ln(coth(x))coth^{2}(x)} + \frac{12log_{coth(x)}^{tanh(x)}csch^{4}(x)}{ln(coth(x))} - \frac{8log_{coth(x)}^{tanh(x)}coth^{2}(x)csch^{2}(x)}{ln(coth(x))}\\ \end{split}\end{equation} \]