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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ log_{cth(x)}^{th(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{cth(x)}^{th(x)}\right)}{dx}\\=&(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})\\=&\frac{-th(x)}{ln(cth(x))} + \frac{1}{ln(cth(x))th(x)} + \frac{log_{cth(x)}^{th(x)}cth(x)}{ln(cth(x))} - \frac{log_{cth(x)}^{th(x)}}{ln(cth(x))cth(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-th(x)}{ln(cth(x))} + \frac{1}{ln(cth(x))th(x)} + \frac{log_{cth(x)}^{th(x)}cth(x)}{ln(cth(x))} - \frac{log_{cth(x)}^{th(x)}}{ln(cth(x))cth(x)}\right)}{dx}\\=&\frac{--(1 - cth^{2}(x))th(x)}{ln^{2}(cth(x))(cth(x))} - \frac{(1 - th^{2}(x))}{ln(cth(x))} + \frac{-(1 - cth^{2}(x))}{ln^{2}(cth(x))(cth(x))th(x)} + \frac{-(1 - th^{2}(x))}{ln(cth(x))th^{2}(x)} + \frac{(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})cth(x)}{ln(cth(x))} + \frac{log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))cth(x)}{ln^{2}(cth(x))(cth(x))} + \frac{log_{cth(x)}^{th(x)}(1 - cth^{2}(x))}{ln(cth(x))} - \frac{(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})}{ln(cth(x))cth(x)} - \frac{log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))}{ln^{2}(cth(x))(cth(x))cth(x)} - \frac{log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))}{ln(cth(x))cth^{2}(x)}\\=&\frac{2th(x)}{ln^{2}(cth(x))cth(x)} - \frac{2th(x)cth(x)}{ln^{2}(cth(x))} - \frac{2}{ln^{2}(cth(x))th(x)cth(x)} + \frac{2cth(x)}{ln^{2}(cth(x))th(x)} - \frac{1}{ln(cth(x))th^{2}(x)} + \frac{th^{2}(x)}{ln(cth(x))} + \frac{2log_{cth(x)}^{th(x)}cth^{2}(x)}{ln^{2}(cth(x))} + \frac{log_{cth(x)}^{th(x)}}{ln(cth(x))cth^{2}(x)} + \frac{2log_{cth(x)}^{th(x)}}{ln^{2}(cth(x))cth^{2}(x)} - \frac{log_{cth(x)}^{th(x)}cth^{2}(x)}{ln(cth(x))} - \frac{4log_{cth(x)}^{th(x)}}{ln^{2}(cth(x))}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2th(x)}{ln^{2}(cth(x))cth(x)} - \frac{2th(x)cth(x)}{ln^{2}(cth(x))} - \frac{2}{ln^{2}(cth(x))th(x)cth(x)} + \frac{2cth(x)}{ln^{2}(cth(x))th(x)} - \frac{1}{ln(cth(x))th^{2}(x)} + \frac{th^{2}(x)}{ln(cth(x))} + \frac{2log_{cth(x)}^{th(x)}cth^{2}(x)}{ln^{2}(cth(x))} + \frac{log_{cth(x)}^{th(x)}}{ln(cth(x))cth^{2}(x)} + \frac{2log_{cth(x)}^{th(x)}}{ln^{2}(cth(x))cth^{2}(x)} - \frac{log_{cth(x)}^{th(x)}cth^{2}(x)}{ln(cth(x))} - \frac{4log_{cth(x)}^{th(x)}}{ln^{2}(cth(x))}\right)}{dx}\\=&\frac{2*-2(1 - cth^{2}(x))th(x)}{ln^{3}(cth(x))(cth(x))cth(x)} + \frac{2(1 - th^{2}(x))}{ln^{2}(cth(x))cth(x)} + \frac{2th(x)*-(1 - cth^{2}(x))}{ln^{2}(cth(x))cth^{2}(x)} - \frac{2*-2(1 - cth^{2}(x))th(x)cth(x)}{ln^{3}(cth(x))(cth(x))} - \frac{2(1 - th^{2}(x))cth(x)}{ln^{2}(cth(x))} - \frac{2th(x)(1 - cth^{2}(x))}{ln^{2}(cth(x))} - \frac{2*-2(1 - cth^{2}(x))}{ln^{3}(cth(x))(cth(x))th(x)cth(x)} - \frac{2*-(1 - th^{2}(x))}{ln^{2}(cth(x))th^{2}(x)cth(x)} - \frac{2*-(1 - cth^{2}(x))}{ln^{2}(cth(x))th(x)cth^{2}(x)} + \frac{2*-2(1 - cth^{2}(x))cth(x)}{ln^{3}(cth(x))(cth(x))th(x)} + \frac{2*-(1 - th^{2}(x))cth(x)}{ln^{2}(cth(x))th^{2}(x)} + \frac{2(1 - cth^{2}(x))}{ln^{2}(cth(x))th(x)} - \frac{-(1 - cth^{2}(x))}{ln^{2}(cth(x))(cth(x))th^{2}(x)} - \frac{-2(1 - th^{2}(x))}{ln(cth(x))th^{3}(x)} + \frac{-(1 - cth^{2}(x))th^{2}(x)}{ln^{2}(cth(x))(cth(x))} + \frac{2th(x)(1 - th^{2}(x))}{ln(cth(x))} + \frac{2(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})cth^{2}(x)}{ln^{2}(cth(x))} + \frac{2log_{cth(x)}^{th(x)}*-2(1 - cth^{2}(x))cth^{2}(x)}{ln^{3}(cth(x))(cth(x))} + \frac{2log_{cth(x)}^{th(x)}*2cth(x)(1 - cth^{2}(x))}{ln^{2}(cth(x))} + \frac{(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})}{ln(cth(x))cth^{2}(x)} + \frac{log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))}{ln^{2}(cth(x))(cth(x))cth^{2}(x)} + \frac{log_{cth(x)}^{th(x)}*-2(1 - cth^{2}(x))}{ln(cth(x))cth^{3}(x)} + \frac{2(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})}{ln^{2}(cth(x))cth^{2}(x)} + \frac{2log_{cth(x)}^{th(x)}*-2(1 - cth^{2}(x))}{ln^{3}(cth(x))(cth(x))cth^{2}(x)} + \frac{2log_{cth(x)}^{th(x)}*-2(1 - cth^{2}(x))}{ln^{2}(cth(x))cth^{3}(x)} - \frac{(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})cth^{2}(x)}{ln(cth(x))} - \frac{log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))cth^{2}(x)}{ln^{2}(cth(x))(cth(x))} - \frac{log_{cth(x)}^{th(x)}*2cth(x)(1 - cth^{2}(x))}{ln(cth(x))} - \frac{4(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})}{ln^{2}(cth(x))} - \frac{4log_{cth(x)}^{th(x)}*-2(1 - cth^{2}(x))}{ln^{3}(cth(x))(cth(x))}\\=&\frac{-6th(x)}{ln^{3}(cth(x))cth^{2}(x)} - \frac{3th(x)}{ln^{2}(cth(x))cth^{2}(x)} - \frac{6th(x)cth^{2}(x)}{ln^{3}(cth(x))} + \frac{3th^{2}(x)cth(x)}{ln^{2}(cth(x))} + \frac{3}{ln^{2}(cth(x))th^{2}(x)cth(x)} + \frac{3th(x)cth^{2}(x)}{ln^{2}(cth(x))} + \frac{6}{ln^{3}(cth(x))th(x)cth^{2}(x)} + \frac{3}{ln^{2}(cth(x))th(x)cth^{2}(x)} - \frac{3cth(x)}{ln^{2}(cth(x))th^{2}(x)} - \frac{3th^{2}(x)}{ln^{2}(cth(x))cth(x)} + \frac{6cth^{2}(x)}{ln^{3}(cth(x))th(x)} - \frac{3cth^{2}(x)}{ln^{2}(cth(x))th(x)} - \frac{12}{ln^{3}(cth(x))th(x)} + \frac{12th(x)}{ln^{3}(cth(x))} + \frac{2}{ln(cth(x))th^{3}(x)} - \frac{2}{ln(cth(x))th(x)} + \frac{2th(x)}{ln(cth(x))} - \frac{2th^{3}(x)}{ln(cth(x))} + \frac{6log_{cth(x)}^{th(x)}cth^{3}(x)}{ln^{3}(cth(x))} + \frac{6log_{cth(x)}^{th(x)}}{ln^{2}(cth(x))cth(x)} - \frac{18log_{cth(x)}^{th(x)}cth(x)}{ln^{3}(cth(x))} + \frac{6log_{cth(x)}^{th(x)}cth(x)}{ln^{2}(cth(x))} - \frac{6log_{cth(x)}^{th(x)}cth^{3}(x)}{ln^{2}(cth(x))} + \frac{18log_{cth(x)}^{th(x)}}{ln^{3}(cth(x))cth(x)} - \frac{6log_{cth(x)}^{th(x)}}{ln^{2}(cth(x))cth^{3}(x)} - \frac{2log_{cth(x)}^{th(x)}}{ln(cth(x))cth^{3}(x)} + \frac{2log_{cth(x)}^{th(x)}}{ln(cth(x))cth(x)} - \frac{6log_{cth(x)}^{th(x)}}{ln^{3}(cth(x))cth^{3}(x)} - \frac{2log_{cth(x)}^{th(x)}cth(x)}{ln(cth(x))} + \frac{2log_{cth(x)}^{th(x)}cth^{3}(x)}{ln(cth(x))}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-6th(x)}{ln^{3}(cth(x))cth^{2}(x)} - \frac{3th(x)}{ln^{2}(cth(x))cth^{2}(x)} - \frac{6th(x)cth^{2}(x)}{ln^{3}(cth(x))} + \frac{3th^{2}(x)cth(x)}{ln^{2}(cth(x))} + \frac{3}{ln^{2}(cth(x))th^{2}(x)cth(x)} + \frac{3th(x)cth^{2}(x)}{ln^{2}(cth(x))} + \frac{6}{ln^{3}(cth(x))th(x)cth^{2}(x)} + \frac{3}{ln^{2}(cth(x))th(x)cth^{2}(x)} - \frac{3cth(x)}{ln^{2}(cth(x))th^{2}(x)} - \frac{3th^{2}(x)}{ln^{2}(cth(x))cth(x)} + \frac{6cth^{2}(x)}{ln^{3}(cth(x))th(x)} - \frac{3cth^{2}(x)}{ln^{2}(cth(x))th(x)} - \frac{12}{ln^{3}(cth(x))th(x)} + \frac{12th(x)}{ln^{3}(cth(x))} + \frac{2}{ln(cth(x))th^{3}(x)} - \frac{2}{ln(cth(x))th(x)} + \frac{2th(x)}{ln(cth(x))} - \frac{2th^{3}(x)}{ln(cth(x))} + \frac{6log_{cth(x)}^{th(x)}cth^{3}(x)}{ln^{3}(cth(x))} + \frac{6log_{cth(x)}^{th(x)}}{ln^{2}(cth(x))cth(x)} - \frac{18log_{cth(x)}^{th(x)}cth(x)}{ln^{3}(cth(x))} + \frac{6log_{cth(x)}^{th(x)}cth(x)}{ln^{2}(cth(x))} - \frac{6log_{cth(x)}^{th(x)}cth^{3}(x)}{ln^{2}(cth(x))} + \frac{18log_{cth(x)}^{th(x)}}{ln^{3}(cth(x))cth(x)} - \frac{6log_{cth(x)}^{th(x)}}{ln^{2}(cth(x))cth^{3}(x)} - \frac{2log_{cth(x)}^{th(x)}}{ln(cth(x))cth^{3}(x)} + \frac{2log_{cth(x)}^{th(x)}}{ln(cth(x))cth(x)} - \frac{6log_{cth(x)}^{th(x)}}{ln^{3}(cth(x))cth^{3}(x)} - \frac{2log_{cth(x)}^{th(x)}cth(x)}{ln(cth(x))} + \frac{2log_{cth(x)}^{th(x)}cth^{3}(x)}{ln(cth(x))}\right)}{dx}\\=&\frac{-6*-3(1 - cth^{2}(x))th(x)}{ln^{4}(cth(x))(cth(x))cth^{2}(x)} - \frac{6(1 - th^{2}(x))}{ln^{3}(cth(x))cth^{2}(x)} - \frac{6th(x)*-2(1 - cth^{2}(x))}{ln^{3}(cth(x))cth^{3}(x)} - \frac{3*-2(1 - cth^{2}(x))th(x)}{ln^{3}(cth(x))(cth(x))cth^{2}(x)} - \frac{3(1 - th^{2}(x))}{ln^{2}(cth(x))cth^{2}(x)} - \frac{3th(x)*-2(1 - cth^{2}(x))}{ln^{2}(cth(x))cth^{3}(x)} - \frac{6*-3(1 - cth^{2}(x))th(x)cth^{2}(x)}{ln^{4}(cth(x))(cth(x))} - \frac{6(1 - th^{2}(x))cth^{2}(x)}{ln^{3}(cth(x))} - \frac{6th(x)*2cth(x)(1 - cth^{2}(x))}{ln^{3}(cth(x))} + \frac{3*-2(1 - cth^{2}(x))th^{2}(x)cth(x)}{ln^{3}(cth(x))(cth(x))} + \frac{3*2th(x)(1 - th^{2}(x))cth(x)}{ln^{2}(cth(x))} + \frac{3th^{2}(x)(1 - cth^{2}(x))}{ln^{2}(cth(x))} + \frac{3*-2(1 - cth^{2}(x))}{ln^{3}(cth(x))(cth(x))th^{2}(x)cth(x)} + \frac{3*-2(1 - th^{2}(x))}{ln^{2}(cth(x))th^{3}(x)cth(x)} + \frac{3*-(1 - cth^{2}(x))}{ln^{2}(cth(x))th^{2}(x)cth^{2}(x)} + \frac{3*-2(1 - cth^{2}(x))th(x)cth^{2}(x)}{ln^{3}(cth(x))(cth(x))} + \frac{3(1 - th^{2}(x))cth^{2}(x)}{ln^{2}(cth(x))} + \frac{3th(x)*2cth(x)(1 - cth^{2}(x))}{ln^{2}(cth(x))} + \frac{6*-3(1 - cth^{2}(x))}{ln^{4}(cth(x))(cth(x))th(x)cth^{2}(x)} + \frac{6*-(1 - th^{2}(x))}{ln^{3}(cth(x))th^{2}(x)cth^{2}(x)} + \frac{6*-2(1 - cth^{2}(x))}{ln^{3}(cth(x))th(x)cth^{3}(x)} + \frac{3*-2(1 - cth^{2}(x))}{ln^{3}(cth(x))(cth(x))th(x)cth^{2}(x)} + \frac{3*-(1 - th^{2}(x))}{ln^{2}(cth(x))th^{2}(x)cth^{2}(x)} + \frac{3*-2(1 - cth^{2}(x))}{ln^{2}(cth(x))th(x)cth^{3}(x)} - \frac{3*-2(1 - cth^{2}(x))cth(x)}{ln^{3}(cth(x))(cth(x))th^{2}(x)} - \frac{3*-2(1 - th^{2}(x))cth(x)}{ln^{2}(cth(x))th^{3}(x)} - \frac{3(1 - cth^{2}(x))}{ln^{2}(cth(x))th^{2}(x)} - \frac{3*-2(1 - cth^{2}(x))th^{2}(x)}{ln^{3}(cth(x))(cth(x))cth(x)} - \frac{3*2th(x)(1 - th^{2}(x))}{ln^{2}(cth(x))cth(x)} - \frac{3th^{2}(x)*-(1 - cth^{2}(x))}{ln^{2}(cth(x))cth^{2}(x)} + \frac{6*-3(1 - cth^{2}(x))cth^{2}(x)}{ln^{4}(cth(x))(cth(x))th(x)} + \frac{6*-(1 - th^{2}(x))cth^{2}(x)}{ln^{3}(cth(x))th^{2}(x)} + \frac{6*2cth(x)(1 - cth^{2}(x))}{ln^{3}(cth(x))th(x)} - \frac{3*-2(1 - cth^{2}(x))cth^{2}(x)}{ln^{3}(cth(x))(cth(x))th(x)} - \frac{3*-(1 - th^{2}(x))cth^{2}(x)}{ln^{2}(cth(x))th^{2}(x)} - \frac{3*2cth(x)(1 - cth^{2}(x))}{ln^{2}(cth(x))th(x)} - \frac{12*-3(1 - cth^{2}(x))}{ln^{4}(cth(x))(cth(x))th(x)} - \frac{12*-(1 - th^{2}(x))}{ln^{3}(cth(x))th^{2}(x)} + \frac{12*-3(1 - cth^{2}(x))th(x)}{ln^{4}(cth(x))(cth(x))} + \frac{12(1 - th^{2}(x))}{ln^{3}(cth(x))} + \frac{2*-(1 - cth^{2}(x))}{ln^{2}(cth(x))(cth(x))th^{3}(x)} + \frac{2*-3(1 - th^{2}(x))}{ln(cth(x))th^{4}(x)} - \frac{2*-(1 - cth^{2}(x))}{ln^{2}(cth(x))(cth(x))th(x)} - \frac{2*-(1 - th^{2}(x))}{ln(cth(x))th^{2}(x)} + \frac{2*-(1 - cth^{2}(x))th(x)}{ln^{2}(cth(x))(cth(x))} + \frac{2(1 - th^{2}(x))}{ln(cth(x))} - \frac{2*-(1 - cth^{2}(x))th^{3}(x)}{ln^{2}(cth(x))(cth(x))} - \frac{2*3th^{2}(x)(1 - th^{2}(x))}{ln(cth(x))} + \frac{6(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})cth^{3}(x)}{ln^{3}(cth(x))} + \frac{6log_{cth(x)}^{th(x)}*-3(1 - cth^{2}(x))cth^{3}(x)}{ln^{4}(cth(x))(cth(x))} + \frac{6log_{cth(x)}^{th(x)}*3cth^{2}(x)(1 - cth^{2}(x))}{ln^{3}(cth(x))} + \frac{6(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})}{ln^{2}(cth(x))cth(x)} + \frac{6log_{cth(x)}^{th(x)}*-2(1 - cth^{2}(x))}{ln^{3}(cth(x))(cth(x))cth(x)} + \frac{6log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))}{ln^{2}(cth(x))cth^{2}(x)} - \frac{18(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})cth(x)}{ln^{3}(cth(x))} - \frac{18log_{cth(x)}^{th(x)}*-3(1 - cth^{2}(x))cth(x)}{ln^{4}(cth(x))(cth(x))} - \frac{18log_{cth(x)}^{th(x)}(1 - cth^{2}(x))}{ln^{3}(cth(x))} + \frac{6(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})cth(x)}{ln^{2}(cth(x))} + \frac{6log_{cth(x)}^{th(x)}*-2(1 - cth^{2}(x))cth(x)}{ln^{3}(cth(x))(cth(x))} + \frac{6log_{cth(x)}^{th(x)}(1 - cth^{2}(x))}{ln^{2}(cth(x))} - \frac{6(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})cth^{3}(x)}{ln^{2}(cth(x))} - \frac{6log_{cth(x)}^{th(x)}*-2(1 - cth^{2}(x))cth^{3}(x)}{ln^{3}(cth(x))(cth(x))} - \frac{6log_{cth(x)}^{th(x)}*3cth^{2}(x)(1 - cth^{2}(x))}{ln^{2}(cth(x))} + \frac{18(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})}{ln^{3}(cth(x))cth(x)} + \frac{18log_{cth(x)}^{th(x)}*-3(1 - cth^{2}(x))}{ln^{4}(cth(x))(cth(x))cth(x)} + \frac{18log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))}{ln^{3}(cth(x))cth^{2}(x)} - \frac{6(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})}{ln^{2}(cth(x))cth^{3}(x)} - \frac{6log_{cth(x)}^{th(x)}*-2(1 - cth^{2}(x))}{ln^{3}(cth(x))(cth(x))cth^{3}(x)} - \frac{6log_{cth(x)}^{th(x)}*-3(1 - cth^{2}(x))}{ln^{2}(cth(x))cth^{4}(x)} - \frac{2(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})}{ln(cth(x))cth^{3}(x)} - \frac{2log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))}{ln^{2}(cth(x))(cth(x))cth^{3}(x)} - \frac{2log_{cth(x)}^{th(x)}*-3(1 - cth^{2}(x))}{ln(cth(x))cth^{4}(x)} + \frac{2(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})}{ln(cth(x))cth(x)} + \frac{2log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))}{ln^{2}(cth(x))(cth(x))cth(x)} + \frac{2log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))}{ln(cth(x))cth^{2}(x)} - \frac{6(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})}{ln^{3}(cth(x))cth^{3}(x)} - \frac{6log_{cth(x)}^{th(x)}*-3(1 - cth^{2}(x))}{ln^{4}(cth(x))(cth(x))cth^{3}(x)} - \frac{6log_{cth(x)}^{th(x)}*-3(1 - cth^{2}(x))}{ln^{3}(cth(x))cth^{4}(x)} - \frac{2(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})cth(x)}{ln(cth(x))} - \frac{2log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))cth(x)}{ln^{2}(cth(x))(cth(x))} - \frac{2log_{cth(x)}^{th(x)}(1 - cth^{2}(x))}{ln(cth(x))} + \frac{2(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})cth^{3}(x)}{ln(cth(x))} + \frac{2log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))cth^{3}(x)}{ln^{2}(cth(x))(cth(x))} + \frac{2log_{cth(x)}^{th(x)}*3cth^{2}(x)(1 - cth^{2}(x))}{ln(cth(x))}\\=&\frac{24th(x)}{ln^{4}(cth(x))cth^{3}(x)} - \frac{72th(x)}{ln^{4}(cth(x))cth(x)} - \frac{24th(x)}{ln^{3}(cth(x))cth(x)} + \frac{24th(x)}{ln^{3}(cth(x))cth^{3}(x)} - \frac{16th(x)}{ln^{2}(cth(x))cth(x)} + \frac{24th(x)cth^{3}(x)}{ln^{3}(cth(x))} + \frac{8th(x)}{ln^{2}(cth(x))cth^{3}(x)} + \frac{72th(x)cth(x)}{ln^{4}(cth(x))} - \frac{24th(x)cth^{3}(x)}{ln^{4}(cth(x))} - \frac{8th(x)cth^{3}(x)}{ln^{2}(cth(x))} - \frac{24th(x)cth(x)}{ln^{3}(cth(x))} + \frac{24}{ln^{3}(cth(x))th(x)cth(x)} + \frac{12th^{2}(x)cth^{2}(x)}{ln^{3}(cth(x))} + \frac{16th(x)cth(x)}{ln^{2}(cth(x))} - \frac{8th^{3}(x)cth(x)}{ln^{2}(cth(x))} - \frac{8}{ln^{2}(cth(x))th^{3}(x)cth(x)} + \frac{16}{ln^{2}(cth(x))th(x)cth(x)} - \frac{12}{ln^{3}(cth(x))th^{2}(x)cth^{2}(x)} - \frac{6}{ln^{2}(cth(x))th^{2}(x)cth^{2}(x)} - \frac{24cth^{3}(x)}{ln^{3}(cth(x))th(x)} + \frac{8cth^{3}(x)}{ln^{2}(cth(x))th(x)} - \frac{24}{ln^{4}(cth(x))th(x)cth^{3}(x)} + \frac{72}{ln^{4}(cth(x))th(x)cth(x)} - \frac{24}{ln^{3}(cth(x))th(x)cth^{3}(x)} + \frac{6th^{2}(x)}{ln^{2}(cth(x))cth^{2}(x)} - \frac{8}{ln^{2}(cth(x))th(x)cth^{3}(x)} + \frac{8cth(x)}{ln^{2}(cth(x))th^{3}(x)} - \frac{6th^{2}(x)cth^{2}(x)}{ln^{2}(cth(x))} - \frac{12cth^{2}(x)}{ln^{3}(cth(x))th^{2}(x)} - \frac{16cth(x)}{ln^{2}(cth(x))th(x)} + \frac{6cth^{2}(x)}{ln^{2}(cth(x))th^{2}(x)} + \frac{12th^{2}(x)}{ln^{3}(cth(x))cth^{2}(x)} + \frac{8th^{3}(x)}{ln^{2}(cth(x))cth(x)} + \frac{24cth(x)}{ln^{3}(cth(x))th(x)} - \frac{72cth(x)}{ln^{4}(cth(x))th(x)} + \frac{24cth^{3}(x)}{ln^{4}(cth(x))th(x)} - \frac{24th^{2}(x)}{ln^{3}(cth(x))} + \frac{24}{ln^{3}(cth(x))th^{2}(x)} - \frac{6}{ln(cth(x))th^{4}(x)} + \frac{8}{ln(cth(x))th^{2}(x)} - \frac{8th^{2}(x)}{ln(cth(x))} + \frac{6th^{4}(x)}{ln(cth(x))} + \frac{24log_{cth(x)}^{th(x)}cth^{4}(x)}{ln^{4}(cth(x))} - \frac{96log_{cth(x)}^{th(x)}cth^{2}(x)}{ln^{4}(cth(x))} + \frac{72log_{cth(x)}^{th(x)}cth^{2}(x)}{ln^{3}(cth(x))} - \frac{36log_{cth(x)}^{th(x)}cth^{4}(x)}{ln^{3}(cth(x))} - \frac{72log_{cth(x)}^{th(x)}}{ln^{3}(cth(x))cth^{2}(x)} - \frac{32log_{cth(x)}^{th(x)}}{ln^{2}(cth(x))cth^{2}(x)} - \frac{32log_{cth(x)}^{th(x)}cth^{2}(x)}{ln^{2}(cth(x))} + \frac{22log_{cth(x)}^{th(x)}cth^{4}(x)}{ln^{2}(cth(x))} + \frac{22log_{cth(x)}^{th(x)}}{ln^{2}(cth(x))cth^{4}(x)} - \frac{96log_{cth(x)}^{th(x)}}{ln^{4}(cth(x))cth^{2}(x)} + \frac{6log_{cth(x)}^{th(x)}}{ln(cth(x))cth^{4}(x)} - \frac{8log_{cth(x)}^{th(x)}}{ln(cth(x))cth^{2}(x)} + \frac{36log_{cth(x)}^{th(x)}}{ln^{3}(cth(x))cth^{4}(x)} + \frac{8log_{cth(x)}^{th(x)}cth^{2}(x)}{ln(cth(x))} - \frac{6log_{cth(x)}^{th(x)}cth^{4}(x)}{ln(cth(x))} + \frac{24log_{cth(x)}^{th(x)}}{ln^{4}(cth(x))cth^{4}(x)} + \frac{144log_{cth(x)}^{th(x)}}{ln^{4}(cth(x))} + \frac{20log_{cth(x)}^{th(x)}}{ln^{2}(cth(x))}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ log_{cth(x)}^{th(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{cth(x)}^{th(x)}\right)}{dx}\\=&(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})\\=&\frac{-th(x)}{ln(cth(x))} + \frac{1}{ln(cth(x))th(x)} + \frac{log_{cth(x)}^{th(x)}cth(x)}{ln(cth(x))} - \frac{log_{cth(x)}^{th(x)}}{ln(cth(x))cth(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-th(x)}{ln(cth(x))} + \frac{1}{ln(cth(x))th(x)} + \frac{log_{cth(x)}^{th(x)}cth(x)}{ln(cth(x))} - \frac{log_{cth(x)}^{th(x)}}{ln(cth(x))cth(x)}\right)}{dx}\\=&\frac{--(1 - cth^{2}(x))th(x)}{ln^{2}(cth(x))(cth(x))} - \frac{(1 - th^{2}(x))}{ln(cth(x))} + \frac{-(1 - cth^{2}(x))}{ln^{2}(cth(x))(cth(x))th(x)} + \frac{-(1 - th^{2}(x))}{ln(cth(x))th^{2}(x)} + \frac{(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})cth(x)}{ln(cth(x))} + \frac{log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))cth(x)}{ln^{2}(cth(x))(cth(x))} + \frac{log_{cth(x)}^{th(x)}(1 - cth^{2}(x))}{ln(cth(x))} - \frac{(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})}{ln(cth(x))cth(x)} - \frac{log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))}{ln^{2}(cth(x))(cth(x))cth(x)} - \frac{log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))}{ln(cth(x))cth^{2}(x)}\\=&\frac{2th(x)}{ln^{2}(cth(x))cth(x)} - \frac{2th(x)cth(x)}{ln^{2}(cth(x))} - \frac{2}{ln^{2}(cth(x))th(x)cth(x)} + \frac{2cth(x)}{ln^{2}(cth(x))th(x)} - \frac{1}{ln(cth(x))th^{2}(x)} + \frac{th^{2}(x)}{ln(cth(x))} + \frac{2log_{cth(x)}^{th(x)}cth^{2}(x)}{ln^{2}(cth(x))} + \frac{log_{cth(x)}^{th(x)}}{ln(cth(x))cth^{2}(x)} + \frac{2log_{cth(x)}^{th(x)}}{ln^{2}(cth(x))cth^{2}(x)} - \frac{log_{cth(x)}^{th(x)}cth^{2}(x)}{ln(cth(x))} - \frac{4log_{cth(x)}^{th(x)}}{ln^{2}(cth(x))}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2th(x)}{ln^{2}(cth(x))cth(x)} - \frac{2th(x)cth(x)}{ln^{2}(cth(x))} - \frac{2}{ln^{2}(cth(x))th(x)cth(x)} + \frac{2cth(x)}{ln^{2}(cth(x))th(x)} - \frac{1}{ln(cth(x))th^{2}(x)} + \frac{th^{2}(x)}{ln(cth(x))} + \frac{2log_{cth(x)}^{th(x)}cth^{2}(x)}{ln^{2}(cth(x))} + \frac{log_{cth(x)}^{th(x)}}{ln(cth(x))cth^{2}(x)} + \frac{2log_{cth(x)}^{th(x)}}{ln^{2}(cth(x))cth^{2}(x)} - \frac{log_{cth(x)}^{th(x)}cth^{2}(x)}{ln(cth(x))} - \frac{4log_{cth(x)}^{th(x)}}{ln^{2}(cth(x))}\right)}{dx}\\=&\frac{2*-2(1 - cth^{2}(x))th(x)}{ln^{3}(cth(x))(cth(x))cth(x)} + \frac{2(1 - th^{2}(x))}{ln^{2}(cth(x))cth(x)} + \frac{2th(x)*-(1 - cth^{2}(x))}{ln^{2}(cth(x))cth^{2}(x)} - \frac{2*-2(1 - cth^{2}(x))th(x)cth(x)}{ln^{3}(cth(x))(cth(x))} - \frac{2(1 - th^{2}(x))cth(x)}{ln^{2}(cth(x))} - \frac{2th(x)(1 - cth^{2}(x))}{ln^{2}(cth(x))} - \frac{2*-2(1 - cth^{2}(x))}{ln^{3}(cth(x))(cth(x))th(x)cth(x)} - \frac{2*-(1 - th^{2}(x))}{ln^{2}(cth(x))th^{2}(x)cth(x)} - \frac{2*-(1 - cth^{2}(x))}{ln^{2}(cth(x))th(x)cth^{2}(x)} + \frac{2*-2(1 - cth^{2}(x))cth(x)}{ln^{3}(cth(x))(cth(x))th(x)} + \frac{2*-(1 - th^{2}(x))cth(x)}{ln^{2}(cth(x))th^{2}(x)} + \frac{2(1 - cth^{2}(x))}{ln^{2}(cth(x))th(x)} - \frac{-(1 - cth^{2}(x))}{ln^{2}(cth(x))(cth(x))th^{2}(x)} - \frac{-2(1 - th^{2}(x))}{ln(cth(x))th^{3}(x)} + \frac{-(1 - cth^{2}(x))th^{2}(x)}{ln^{2}(cth(x))(cth(x))} + \frac{2th(x)(1 - th^{2}(x))}{ln(cth(x))} + \frac{2(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})cth^{2}(x)}{ln^{2}(cth(x))} + \frac{2log_{cth(x)}^{th(x)}*-2(1 - cth^{2}(x))cth^{2}(x)}{ln^{3}(cth(x))(cth(x))} + \frac{2log_{cth(x)}^{th(x)}*2cth(x)(1 - cth^{2}(x))}{ln^{2}(cth(x))} + \frac{(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})}{ln(cth(x))cth^{2}(x)} + \frac{log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))}{ln^{2}(cth(x))(cth(x))cth^{2}(x)} + \frac{log_{cth(x)}^{th(x)}*-2(1 - cth^{2}(x))}{ln(cth(x))cth^{3}(x)} + \frac{2(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})}{ln^{2}(cth(x))cth^{2}(x)} + \frac{2log_{cth(x)}^{th(x)}*-2(1 - cth^{2}(x))}{ln^{3}(cth(x))(cth(x))cth^{2}(x)} + \frac{2log_{cth(x)}^{th(x)}*-2(1 - cth^{2}(x))}{ln^{2}(cth(x))cth^{3}(x)} - \frac{(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})cth^{2}(x)}{ln(cth(x))} - \frac{log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))cth^{2}(x)}{ln^{2}(cth(x))(cth(x))} - \frac{log_{cth(x)}^{th(x)}*2cth(x)(1 - cth^{2}(x))}{ln(cth(x))} - \frac{4(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})}{ln^{2}(cth(x))} - \frac{4log_{cth(x)}^{th(x)}*-2(1 - cth^{2}(x))}{ln^{3}(cth(x))(cth(x))}\\=&\frac{-6th(x)}{ln^{3}(cth(x))cth^{2}(x)} - \frac{3th(x)}{ln^{2}(cth(x))cth^{2}(x)} - \frac{6th(x)cth^{2}(x)}{ln^{3}(cth(x))} + \frac{3th^{2}(x)cth(x)}{ln^{2}(cth(x))} + \frac{3}{ln^{2}(cth(x))th^{2}(x)cth(x)} + \frac{3th(x)cth^{2}(x)}{ln^{2}(cth(x))} + \frac{6}{ln^{3}(cth(x))th(x)cth^{2}(x)} + \frac{3}{ln^{2}(cth(x))th(x)cth^{2}(x)} - \frac{3cth(x)}{ln^{2}(cth(x))th^{2}(x)} - \frac{3th^{2}(x)}{ln^{2}(cth(x))cth(x)} + \frac{6cth^{2}(x)}{ln^{3}(cth(x))th(x)} - \frac{3cth^{2}(x)}{ln^{2}(cth(x))th(x)} - \frac{12}{ln^{3}(cth(x))th(x)} + \frac{12th(x)}{ln^{3}(cth(x))} + \frac{2}{ln(cth(x))th^{3}(x)} - \frac{2}{ln(cth(x))th(x)} + \frac{2th(x)}{ln(cth(x))} - \frac{2th^{3}(x)}{ln(cth(x))} + \frac{6log_{cth(x)}^{th(x)}cth^{3}(x)}{ln^{3}(cth(x))} + \frac{6log_{cth(x)}^{th(x)}}{ln^{2}(cth(x))cth(x)} - \frac{18log_{cth(x)}^{th(x)}cth(x)}{ln^{3}(cth(x))} + \frac{6log_{cth(x)}^{th(x)}cth(x)}{ln^{2}(cth(x))} - \frac{6log_{cth(x)}^{th(x)}cth^{3}(x)}{ln^{2}(cth(x))} + \frac{18log_{cth(x)}^{th(x)}}{ln^{3}(cth(x))cth(x)} - \frac{6log_{cth(x)}^{th(x)}}{ln^{2}(cth(x))cth^{3}(x)} - \frac{2log_{cth(x)}^{th(x)}}{ln(cth(x))cth^{3}(x)} + \frac{2log_{cth(x)}^{th(x)}}{ln(cth(x))cth(x)} - \frac{6log_{cth(x)}^{th(x)}}{ln^{3}(cth(x))cth^{3}(x)} - \frac{2log_{cth(x)}^{th(x)}cth(x)}{ln(cth(x))} + \frac{2log_{cth(x)}^{th(x)}cth^{3}(x)}{ln(cth(x))}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-6th(x)}{ln^{3}(cth(x))cth^{2}(x)} - \frac{3th(x)}{ln^{2}(cth(x))cth^{2}(x)} - \frac{6th(x)cth^{2}(x)}{ln^{3}(cth(x))} + \frac{3th^{2}(x)cth(x)}{ln^{2}(cth(x))} + \frac{3}{ln^{2}(cth(x))th^{2}(x)cth(x)} + \frac{3th(x)cth^{2}(x)}{ln^{2}(cth(x))} + \frac{6}{ln^{3}(cth(x))th(x)cth^{2}(x)} + \frac{3}{ln^{2}(cth(x))th(x)cth^{2}(x)} - \frac{3cth(x)}{ln^{2}(cth(x))th^{2}(x)} - \frac{3th^{2}(x)}{ln^{2}(cth(x))cth(x)} + \frac{6cth^{2}(x)}{ln^{3}(cth(x))th(x)} - \frac{3cth^{2}(x)}{ln^{2}(cth(x))th(x)} - \frac{12}{ln^{3}(cth(x))th(x)} + \frac{12th(x)}{ln^{3}(cth(x))} + \frac{2}{ln(cth(x))th^{3}(x)} - \frac{2}{ln(cth(x))th(x)} + \frac{2th(x)}{ln(cth(x))} - \frac{2th^{3}(x)}{ln(cth(x))} + \frac{6log_{cth(x)}^{th(x)}cth^{3}(x)}{ln^{3}(cth(x))} + \frac{6log_{cth(x)}^{th(x)}}{ln^{2}(cth(x))cth(x)} - \frac{18log_{cth(x)}^{th(x)}cth(x)}{ln^{3}(cth(x))} + \frac{6log_{cth(x)}^{th(x)}cth(x)}{ln^{2}(cth(x))} - \frac{6log_{cth(x)}^{th(x)}cth^{3}(x)}{ln^{2}(cth(x))} + \frac{18log_{cth(x)}^{th(x)}}{ln^{3}(cth(x))cth(x)} - \frac{6log_{cth(x)}^{th(x)}}{ln^{2}(cth(x))cth^{3}(x)} - \frac{2log_{cth(x)}^{th(x)}}{ln(cth(x))cth^{3}(x)} + \frac{2log_{cth(x)}^{th(x)}}{ln(cth(x))cth(x)} - \frac{6log_{cth(x)}^{th(x)}}{ln^{3}(cth(x))cth^{3}(x)} - \frac{2log_{cth(x)}^{th(x)}cth(x)}{ln(cth(x))} + \frac{2log_{cth(x)}^{th(x)}cth^{3}(x)}{ln(cth(x))}\right)}{dx}\\=&\frac{-6*-3(1 - cth^{2}(x))th(x)}{ln^{4}(cth(x))(cth(x))cth^{2}(x)} - \frac{6(1 - th^{2}(x))}{ln^{3}(cth(x))cth^{2}(x)} - \frac{6th(x)*-2(1 - cth^{2}(x))}{ln^{3}(cth(x))cth^{3}(x)} - \frac{3*-2(1 - cth^{2}(x))th(x)}{ln^{3}(cth(x))(cth(x))cth^{2}(x)} - \frac{3(1 - th^{2}(x))}{ln^{2}(cth(x))cth^{2}(x)} - \frac{3th(x)*-2(1 - cth^{2}(x))}{ln^{2}(cth(x))cth^{3}(x)} - \frac{6*-3(1 - cth^{2}(x))th(x)cth^{2}(x)}{ln^{4}(cth(x))(cth(x))} - \frac{6(1 - th^{2}(x))cth^{2}(x)}{ln^{3}(cth(x))} - \frac{6th(x)*2cth(x)(1 - cth^{2}(x))}{ln^{3}(cth(x))} + \frac{3*-2(1 - cth^{2}(x))th^{2}(x)cth(x)}{ln^{3}(cth(x))(cth(x))} + \frac{3*2th(x)(1 - th^{2}(x))cth(x)}{ln^{2}(cth(x))} + \frac{3th^{2}(x)(1 - cth^{2}(x))}{ln^{2}(cth(x))} + \frac{3*-2(1 - cth^{2}(x))}{ln^{3}(cth(x))(cth(x))th^{2}(x)cth(x)} + \frac{3*-2(1 - th^{2}(x))}{ln^{2}(cth(x))th^{3}(x)cth(x)} + \frac{3*-(1 - cth^{2}(x))}{ln^{2}(cth(x))th^{2}(x)cth^{2}(x)} + \frac{3*-2(1 - cth^{2}(x))th(x)cth^{2}(x)}{ln^{3}(cth(x))(cth(x))} + \frac{3(1 - th^{2}(x))cth^{2}(x)}{ln^{2}(cth(x))} + \frac{3th(x)*2cth(x)(1 - cth^{2}(x))}{ln^{2}(cth(x))} + \frac{6*-3(1 - cth^{2}(x))}{ln^{4}(cth(x))(cth(x))th(x)cth^{2}(x)} + \frac{6*-(1 - th^{2}(x))}{ln^{3}(cth(x))th^{2}(x)cth^{2}(x)} + \frac{6*-2(1 - cth^{2}(x))}{ln^{3}(cth(x))th(x)cth^{3}(x)} + \frac{3*-2(1 - cth^{2}(x))}{ln^{3}(cth(x))(cth(x))th(x)cth^{2}(x)} + \frac{3*-(1 - th^{2}(x))}{ln^{2}(cth(x))th^{2}(x)cth^{2}(x)} + \frac{3*-2(1 - cth^{2}(x))}{ln^{2}(cth(x))th(x)cth^{3}(x)} - \frac{3*-2(1 - cth^{2}(x))cth(x)}{ln^{3}(cth(x))(cth(x))th^{2}(x)} - \frac{3*-2(1 - th^{2}(x))cth(x)}{ln^{2}(cth(x))th^{3}(x)} - \frac{3(1 - cth^{2}(x))}{ln^{2}(cth(x))th^{2}(x)} - \frac{3*-2(1 - cth^{2}(x))th^{2}(x)}{ln^{3}(cth(x))(cth(x))cth(x)} - \frac{3*2th(x)(1 - th^{2}(x))}{ln^{2}(cth(x))cth(x)} - \frac{3th^{2}(x)*-(1 - cth^{2}(x))}{ln^{2}(cth(x))cth^{2}(x)} + \frac{6*-3(1 - cth^{2}(x))cth^{2}(x)}{ln^{4}(cth(x))(cth(x))th(x)} + \frac{6*-(1 - th^{2}(x))cth^{2}(x)}{ln^{3}(cth(x))th^{2}(x)} + \frac{6*2cth(x)(1 - cth^{2}(x))}{ln^{3}(cth(x))th(x)} - \frac{3*-2(1 - cth^{2}(x))cth^{2}(x)}{ln^{3}(cth(x))(cth(x))th(x)} - \frac{3*-(1 - th^{2}(x))cth^{2}(x)}{ln^{2}(cth(x))th^{2}(x)} - \frac{3*2cth(x)(1 - cth^{2}(x))}{ln^{2}(cth(x))th(x)} - \frac{12*-3(1 - cth^{2}(x))}{ln^{4}(cth(x))(cth(x))th(x)} - \frac{12*-(1 - th^{2}(x))}{ln^{3}(cth(x))th^{2}(x)} + \frac{12*-3(1 - cth^{2}(x))th(x)}{ln^{4}(cth(x))(cth(x))} + \frac{12(1 - th^{2}(x))}{ln^{3}(cth(x))} + \frac{2*-(1 - cth^{2}(x))}{ln^{2}(cth(x))(cth(x))th^{3}(x)} + \frac{2*-3(1 - th^{2}(x))}{ln(cth(x))th^{4}(x)} - \frac{2*-(1 - cth^{2}(x))}{ln^{2}(cth(x))(cth(x))th(x)} - \frac{2*-(1 - th^{2}(x))}{ln(cth(x))th^{2}(x)} + \frac{2*-(1 - cth^{2}(x))th(x)}{ln^{2}(cth(x))(cth(x))} + \frac{2(1 - th^{2}(x))}{ln(cth(x))} - \frac{2*-(1 - cth^{2}(x))th^{3}(x)}{ln^{2}(cth(x))(cth(x))} - \frac{2*3th^{2}(x)(1 - th^{2}(x))}{ln(cth(x))} + \frac{6(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})cth^{3}(x)}{ln^{3}(cth(x))} + \frac{6log_{cth(x)}^{th(x)}*-3(1 - cth^{2}(x))cth^{3}(x)}{ln^{4}(cth(x))(cth(x))} + \frac{6log_{cth(x)}^{th(x)}*3cth^{2}(x)(1 - cth^{2}(x))}{ln^{3}(cth(x))} + \frac{6(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})}{ln^{2}(cth(x))cth(x)} + \frac{6log_{cth(x)}^{th(x)}*-2(1 - cth^{2}(x))}{ln^{3}(cth(x))(cth(x))cth(x)} + \frac{6log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))}{ln^{2}(cth(x))cth^{2}(x)} - \frac{18(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})cth(x)}{ln^{3}(cth(x))} - \frac{18log_{cth(x)}^{th(x)}*-3(1 - cth^{2}(x))cth(x)}{ln^{4}(cth(x))(cth(x))} - \frac{18log_{cth(x)}^{th(x)}(1 - cth^{2}(x))}{ln^{3}(cth(x))} + \frac{6(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})cth(x)}{ln^{2}(cth(x))} + \frac{6log_{cth(x)}^{th(x)}*-2(1 - cth^{2}(x))cth(x)}{ln^{3}(cth(x))(cth(x))} + \frac{6log_{cth(x)}^{th(x)}(1 - cth^{2}(x))}{ln^{2}(cth(x))} - \frac{6(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})cth^{3}(x)}{ln^{2}(cth(x))} - \frac{6log_{cth(x)}^{th(x)}*-2(1 - cth^{2}(x))cth^{3}(x)}{ln^{3}(cth(x))(cth(x))} - \frac{6log_{cth(x)}^{th(x)}*3cth^{2}(x)(1 - cth^{2}(x))}{ln^{2}(cth(x))} + \frac{18(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})}{ln^{3}(cth(x))cth(x)} + \frac{18log_{cth(x)}^{th(x)}*-3(1 - cth^{2}(x))}{ln^{4}(cth(x))(cth(x))cth(x)} + \frac{18log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))}{ln^{3}(cth(x))cth^{2}(x)} - \frac{6(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})}{ln^{2}(cth(x))cth^{3}(x)} - \frac{6log_{cth(x)}^{th(x)}*-2(1 - cth^{2}(x))}{ln^{3}(cth(x))(cth(x))cth^{3}(x)} - \frac{6log_{cth(x)}^{th(x)}*-3(1 - cth^{2}(x))}{ln^{2}(cth(x))cth^{4}(x)} - \frac{2(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})}{ln(cth(x))cth^{3}(x)} - \frac{2log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))}{ln^{2}(cth(x))(cth(x))cth^{3}(x)} - \frac{2log_{cth(x)}^{th(x)}*-3(1 - cth^{2}(x))}{ln(cth(x))cth^{4}(x)} + \frac{2(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})}{ln(cth(x))cth(x)} + \frac{2log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))}{ln^{2}(cth(x))(cth(x))cth(x)} + \frac{2log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))}{ln(cth(x))cth^{2}(x)} - \frac{6(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})}{ln^{3}(cth(x))cth^{3}(x)} - \frac{6log_{cth(x)}^{th(x)}*-3(1 - cth^{2}(x))}{ln^{4}(cth(x))(cth(x))cth^{3}(x)} - \frac{6log_{cth(x)}^{th(x)}*-3(1 - cth^{2}(x))}{ln^{3}(cth(x))cth^{4}(x)} - \frac{2(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})cth(x)}{ln(cth(x))} - \frac{2log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))cth(x)}{ln^{2}(cth(x))(cth(x))} - \frac{2log_{cth(x)}^{th(x)}(1 - cth^{2}(x))}{ln(cth(x))} + \frac{2(\frac{(\frac{((1 - th^{2}(x)))}{(th(x))} - \frac{((1 - cth^{2}(x)))log_{cth(x)}^{th(x)}}{(cth(x))})}{(ln(cth(x)))})cth^{3}(x)}{ln(cth(x))} + \frac{2log_{cth(x)}^{th(x)}*-(1 - cth^{2}(x))cth^{3}(x)}{ln^{2}(cth(x))(cth(x))} + \frac{2log_{cth(x)}^{th(x)}*3cth^{2}(x)(1 - cth^{2}(x))}{ln(cth(x))}\\=&\frac{24th(x)}{ln^{4}(cth(x))cth^{3}(x)} - \frac{72th(x)}{ln^{4}(cth(x))cth(x)} - \frac{24th(x)}{ln^{3}(cth(x))cth(x)} + \frac{24th(x)}{ln^{3}(cth(x))cth^{3}(x)} - \frac{16th(x)}{ln^{2}(cth(x))cth(x)} + \frac{24th(x)cth^{3}(x)}{ln^{3}(cth(x))} + \frac{8th(x)}{ln^{2}(cth(x))cth^{3}(x)} + \frac{72th(x)cth(x)}{ln^{4}(cth(x))} - \frac{24th(x)cth^{3}(x)}{ln^{4}(cth(x))} - \frac{8th(x)cth^{3}(x)}{ln^{2}(cth(x))} - \frac{24th(x)cth(x)}{ln^{3}(cth(x))} + \frac{24}{ln^{3}(cth(x))th(x)cth(x)} + \frac{12th^{2}(x)cth^{2}(x)}{ln^{3}(cth(x))} + \frac{16th(x)cth(x)}{ln^{2}(cth(x))} - \frac{8th^{3}(x)cth(x)}{ln^{2}(cth(x))} - \frac{8}{ln^{2}(cth(x))th^{3}(x)cth(x)} + \frac{16}{ln^{2}(cth(x))th(x)cth(x)} - \frac{12}{ln^{3}(cth(x))th^{2}(x)cth^{2}(x)} - \frac{6}{ln^{2}(cth(x))th^{2}(x)cth^{2}(x)} - \frac{24cth^{3}(x)}{ln^{3}(cth(x))th(x)} + \frac{8cth^{3}(x)}{ln^{2}(cth(x))th(x)} - \frac{24}{ln^{4}(cth(x))th(x)cth^{3}(x)} + \frac{72}{ln^{4}(cth(x))th(x)cth(x)} - \frac{24}{ln^{3}(cth(x))th(x)cth^{3}(x)} + \frac{6th^{2}(x)}{ln^{2}(cth(x))cth^{2}(x)} - \frac{8}{ln^{2}(cth(x))th(x)cth^{3}(x)} + \frac{8cth(x)}{ln^{2}(cth(x))th^{3}(x)} - \frac{6th^{2}(x)cth^{2}(x)}{ln^{2}(cth(x))} - \frac{12cth^{2}(x)}{ln^{3}(cth(x))th^{2}(x)} - \frac{16cth(x)}{ln^{2}(cth(x))th(x)} + \frac{6cth^{2}(x)}{ln^{2}(cth(x))th^{2}(x)} + \frac{12th^{2}(x)}{ln^{3}(cth(x))cth^{2}(x)} + \frac{8th^{3}(x)}{ln^{2}(cth(x))cth(x)} + \frac{24cth(x)}{ln^{3}(cth(x))th(x)} - \frac{72cth(x)}{ln^{4}(cth(x))th(x)} + \frac{24cth^{3}(x)}{ln^{4}(cth(x))th(x)} - \frac{24th^{2}(x)}{ln^{3}(cth(x))} + \frac{24}{ln^{3}(cth(x))th^{2}(x)} - \frac{6}{ln(cth(x))th^{4}(x)} + \frac{8}{ln(cth(x))th^{2}(x)} - \frac{8th^{2}(x)}{ln(cth(x))} + \frac{6th^{4}(x)}{ln(cth(x))} + \frac{24log_{cth(x)}^{th(x)}cth^{4}(x)}{ln^{4}(cth(x))} - \frac{96log_{cth(x)}^{th(x)}cth^{2}(x)}{ln^{4}(cth(x))} + \frac{72log_{cth(x)}^{th(x)}cth^{2}(x)}{ln^{3}(cth(x))} - \frac{36log_{cth(x)}^{th(x)}cth^{4}(x)}{ln^{3}(cth(x))} - \frac{72log_{cth(x)}^{th(x)}}{ln^{3}(cth(x))cth^{2}(x)} - \frac{32log_{cth(x)}^{th(x)}}{ln^{2}(cth(x))cth^{2}(x)} - \frac{32log_{cth(x)}^{th(x)}cth^{2}(x)}{ln^{2}(cth(x))} + \frac{22log_{cth(x)}^{th(x)}cth^{4}(x)}{ln^{2}(cth(x))} + \frac{22log_{cth(x)}^{th(x)}}{ln^{2}(cth(x))cth^{4}(x)} - \frac{96log_{cth(x)}^{th(x)}}{ln^{4}(cth(x))cth^{2}(x)} + \frac{6log_{cth(x)}^{th(x)}}{ln(cth(x))cth^{4}(x)} - \frac{8log_{cth(x)}^{th(x)}}{ln(cth(x))cth^{2}(x)} + \frac{36log_{cth(x)}^{th(x)}}{ln^{3}(cth(x))cth^{4}(x)} + \frac{8log_{cth(x)}^{th(x)}cth^{2}(x)}{ln(cth(x))} - \frac{6log_{cth(x)}^{th(x)}cth^{4}(x)}{ln(cth(x))} + \frac{24log_{cth(x)}^{th(x)}}{ln^{4}(cth(x))cth^{4}(x)} + \frac{144log_{cth(x)}^{th(x)}}{ln^{4}(cth(x))} + \frac{20log_{cth(x)}^{th(x)}}{ln^{2}(cth(x))}\\ \end{split}\end{equation} \]