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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ lg(lg(\frac{sin(x)}{sinh(x)}))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( lg(lg(\frac{sin(x)}{sinh(x)}))\right)}{dx}\\=&\frac{(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})}{ln{10}(lg(\frac{sin(x)}{sinh(x)}))ln{10}(\frac{sin(x)}{sinh(x)})}\\=&\frac{cos(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin(x)} - \frac{cosh(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{cos(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin(x)} - \frac{cosh(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh(x)}\right)}{dx}\\=&\frac{-2*0cos(x)}{ln^{3}{10}lg(\frac{sin(x)}{sinh(x)})sin(x)} + \frac{-(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos(x)}{ln^{2}{10}lg^{2}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin(x)} + \frac{-cos(x)cos(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{2}(x)} + \frac{-sin(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin(x)} - \frac{-2*0cosh(x)}{ln^{3}{10}lg(\frac{sin(x)}{sinh(x)})sinh(x)} - \frac{-(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cosh(x)}{ln^{2}{10}lg^{2}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sinh(x)} - \frac{-cosh(x)cosh(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} - \frac{sinh(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh(x)}\\=&\frac{2cos(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh(x)} - \frac{cos^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)} - \frac{cos^{2}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{2}(x)} + \frac{cosh^{2}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} - \frac{cosh^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} - \frac{2}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2cos(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh(x)} - \frac{cos^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)} - \frac{cos^{2}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{2}(x)} + \frac{cosh^{2}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} - \frac{cosh^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} - \frac{2}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})}\right)}{dx}\\=&\frac{2*-3*0cos(x)cosh(x)}{ln^{4}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh(x)} + \frac{2*-2(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos(x)cosh(x)}{ln^{3}{10}lg^{3}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin(x)sinh(x)} + \frac{2*-cos(x)cos(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} + \frac{2*-sin(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh(x)} + \frac{2cos(x)*-cosh(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} + \frac{2cos(x)sinh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh(x)} - \frac{-3*0cos^{2}(x)}{ln^{4}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)} - \frac{-2(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos^{2}(x)}{ln^{3}{10}lg^{3}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin^{2}(x)} - \frac{-2cos(x)cos^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} - \frac{-2cos(x)sin(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)} - \frac{-2*0cos^{2}(x)}{ln^{3}{10}lg(\frac{sin(x)}{sinh(x)})sin^{2}(x)} - \frac{-(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos^{2}(x)}{ln^{2}{10}lg^{2}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin^{2}(x)} - \frac{-2cos(x)cos^{2}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{3}(x)} - \frac{-2cos(x)sin(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{2}(x)} + \frac{-2*0cosh^{2}(x)}{ln^{3}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} + \frac{-(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cosh^{2}(x)}{ln^{2}{10}lg^{2}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} + \frac{-2cosh(x)cosh^{2}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} + \frac{2cosh(x)sinh(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} - \frac{-3*0cosh^{2}(x)}{ln^{4}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} - \frac{-2(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cosh^{2}(x)}{ln^{3}{10}lg^{3}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} - \frac{-2cosh(x)cosh^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} - \frac{2cosh(x)sinh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} - \frac{2*-2*0}{ln^{3}{10}lg(\frac{sin(x)}{sinh(x)})} - \frac{2*-(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})}{ln^{2}{10}lg^{2}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})}\\=&\frac{-6cos^{2}(x)cosh(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} + \frac{6cos(x)cosh^{2}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} - \frac{3cos^{2}(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} - \frac{6cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh(x)} - \frac{3cos(x)cosh^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} + \frac{6cos(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)} + \frac{3cos^{3}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2cos^{3}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2cos^{3}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2cos(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin(x)} - \frac{2cosh^{3}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} + \frac{3cosh^{3}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} + \frac{2cosh(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh(x)} - \frac{2cosh^{3}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-6cos^{2}(x)cosh(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} + \frac{6cos(x)cosh^{2}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} - \frac{3cos^{2}(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} - \frac{6cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh(x)} - \frac{3cos(x)cosh^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} + \frac{6cos(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)} + \frac{3cos^{3}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2cos^{3}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2cos^{3}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2cos(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin(x)} - \frac{2cosh^{3}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} + \frac{3cosh^{3}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} + \frac{2cosh(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh(x)} - \frac{2cosh^{3}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)}\right)}{dx}\\=&\frac{-6*-4*0cos^{2}(x)cosh(x)}{ln^{5}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} - \frac{6*-3(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos^{2}(x)cosh(x)}{ln^{4}{10}lg^{4}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} - \frac{6*-2cos(x)cos^{2}(x)cosh(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{3}(x)sinh(x)} - \frac{6*-2cos(x)sin(x)cosh(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} - \frac{6cos^{2}(x)*-cosh(x)cosh(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh^{2}(x)} - \frac{6cos^{2}(x)sinh(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} + \frac{6*-4*0cos(x)cosh^{2}(x)}{ln^{5}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} + \frac{6*-3(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos(x)cosh^{2}(x)}{ln^{4}{10}lg^{4}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} + \frac{6*-cos(x)cos(x)cosh^{2}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh^{2}(x)} + \frac{6*-sin(x)cosh^{2}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} + \frac{6cos(x)*-2cosh(x)cosh^{2}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{3}(x)} + \frac{6cos(x)*2cosh(x)sinh(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} - \frac{3*-3*0cos^{2}(x)cosh(x)}{ln^{4}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} - \frac{3*-2(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos^{2}(x)cosh(x)}{ln^{3}{10}lg^{3}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} - \frac{3*-2cos(x)cos^{2}(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{3}(x)sinh(x)} - \frac{3*-2cos(x)sin(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} - \frac{3cos^{2}(x)*-cosh(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh^{2}(x)} - \frac{3cos^{2}(x)sinh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} - \frac{6*-3*0cosh(x)}{ln^{4}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh(x)} - \frac{6*-2(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cosh(x)}{ln^{3}{10}lg^{3}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sinh(x)} - \frac{6*-cosh(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} - \frac{6sinh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh(x)} - \frac{3*-3*0cos(x)cosh^{2}(x)}{ln^{4}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} - \frac{3*-2(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos(x)cosh^{2}(x)}{ln^{3}{10}lg^{3}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} - \frac{3*-cos(x)cos(x)cosh^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh^{2}(x)} - \frac{3*-sin(x)cosh^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} - \frac{3cos(x)*-2cosh(x)cosh^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{3}(x)} - \frac{3cos(x)*2cosh(x)sinh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} + \frac{6*-3*0cos(x)}{ln^{4}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)} + \frac{6*-2(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos(x)}{ln^{3}{10}lg^{3}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin(x)} + \frac{6*-cos(x)cos(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)} + \frac{6*-sin(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)} + \frac{3*-3*0cos^{3}(x)}{ln^{4}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{3*-2(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos^{3}(x)}{ln^{3}{10}lg^{3}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{3*-3cos(x)cos^{3}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{4}(x)} + \frac{3*-3cos^{2}(x)sin(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2*-4*0cos^{3}(x)}{ln^{5}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2*-3(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos^{3}(x)}{ln^{4}{10}lg^{4}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2*-3cos(x)cos^{3}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{4}(x)} + \frac{2*-3cos^{2}(x)sin(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2*-2*0cos^{3}(x)}{ln^{3}{10}lg(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2*-(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos^{3}(x)}{ln^{2}{10}lg^{2}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2*-3cos(x)cos^{3}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{4}(x)} + \frac{2*-3cos^{2}(x)sin(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2*-2*0cos(x)}{ln^{3}{10}lg(\frac{sin(x)}{sinh(x)})sin(x)} + \frac{2*-(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos(x)}{ln^{2}{10}lg^{2}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin(x)} + \frac{2*-cos(x)cos(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{2}(x)} + \frac{2*-sin(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin(x)} - \frac{2*-2*0cosh^{3}(x)}{ln^{3}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} - \frac{2*-(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cosh^{3}(x)}{ln^{2}{10}lg^{2}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} - \frac{2*-3cosh(x)cosh^{3}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{4}(x)} - \frac{2*3cosh^{2}(x)sinh(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} + \frac{3*-3*0cosh^{3}(x)}{ln^{4}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} + \frac{3*-2(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cosh^{3}(x)}{ln^{3}{10}lg^{3}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} + \frac{3*-3cosh(x)cosh^{3}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{4}(x)} + \frac{3*3cosh^{2}(x)sinh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} + \frac{2*-2*0cosh(x)}{ln^{3}{10}lg(\frac{sin(x)}{sinh(x)})sinh(x)} + \frac{2*-(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cosh(x)}{ln^{2}{10}lg^{2}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sinh(x)} + \frac{2*-cosh(x)cosh(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} + \frac{2sinh(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh(x)} - \frac{2*-4*0cosh^{3}(x)}{ln^{5}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} - \frac{2*-3(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cosh^{3}(x)}{ln^{4}{10}lg^{4}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} - \frac{2*-3cosh(x)cosh^{3}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sinh^{4}(x)} - \frac{2*3cosh^{2}(x)sinh(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)}\\=&\frac{24cos^{3}(x)cosh(x)}{ln^{5}{10}lg^{4}(\frac{sin(x)}{sinh(x)})sin^{3}(x)sinh(x)} - \frac{36cos^{2}(x)cosh^{2}(x)}{ln^{5}{10}lg^{4}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh^{2}(x)} + \frac{24cos^{3}(x)cosh(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{3}(x)sinh(x)} + \frac{48cos(x)cosh(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin(x)sinh(x)} - \frac{24cos(x)cosh^{3}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{3}(x)} + \frac{24cos(x)cosh^{3}(x)}{ln^{5}{10}lg^{4}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{3}(x)} - \frac{24cosh^{2}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} + \frac{6cos^{2}(x)cosh^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh^{2}(x)} + \frac{8cos(x)cosh^{3}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{3}(x)} + \frac{8cos^{3}(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{3}(x)sinh(x)} - \frac{20cos^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)} + \frac{20cosh^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} + \frac{6cosh^{4}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{4}(x)} - \frac{8cosh^{2}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} - \frac{24cos^{2}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{2}(x)} - \frac{11cos^{4}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{4}(x)} - \frac{12cos^{4}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{4}(x)} - \frac{6cos^{4}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{4}(x)} - \frac{8cos^{2}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{2}(x)} - \frac{6cos^{4}(x)}{ln^{5}{10}lg^{4}(\frac{sin(x)}{sinh(x)})sin^{4}(x)} - \frac{11cosh^{4}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{4}(x)} + \frac{12cosh^{4}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sinh^{4}(x)} - \frac{6cosh^{4}(x)}{ln^{5}{10}lg^{4}(\frac{sin(x)}{sinh(x)})sinh^{4}(x)} - \frac{12}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ lg(lg(\frac{sin(x)}{sinh(x)}))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( lg(lg(\frac{sin(x)}{sinh(x)}))\right)}{dx}\\=&\frac{(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})}{ln{10}(lg(\frac{sin(x)}{sinh(x)}))ln{10}(\frac{sin(x)}{sinh(x)})}\\=&\frac{cos(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin(x)} - \frac{cosh(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{cos(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin(x)} - \frac{cosh(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh(x)}\right)}{dx}\\=&\frac{-2*0cos(x)}{ln^{3}{10}lg(\frac{sin(x)}{sinh(x)})sin(x)} + \frac{-(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos(x)}{ln^{2}{10}lg^{2}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin(x)} + \frac{-cos(x)cos(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{2}(x)} + \frac{-sin(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin(x)} - \frac{-2*0cosh(x)}{ln^{3}{10}lg(\frac{sin(x)}{sinh(x)})sinh(x)} - \frac{-(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cosh(x)}{ln^{2}{10}lg^{2}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sinh(x)} - \frac{-cosh(x)cosh(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} - \frac{sinh(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh(x)}\\=&\frac{2cos(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh(x)} - \frac{cos^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)} - \frac{cos^{2}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{2}(x)} + \frac{cosh^{2}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} - \frac{cosh^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} - \frac{2}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2cos(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh(x)} - \frac{cos^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)} - \frac{cos^{2}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{2}(x)} + \frac{cosh^{2}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} - \frac{cosh^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} - \frac{2}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})}\right)}{dx}\\=&\frac{2*-3*0cos(x)cosh(x)}{ln^{4}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh(x)} + \frac{2*-2(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos(x)cosh(x)}{ln^{3}{10}lg^{3}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin(x)sinh(x)} + \frac{2*-cos(x)cos(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} + \frac{2*-sin(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh(x)} + \frac{2cos(x)*-cosh(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} + \frac{2cos(x)sinh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh(x)} - \frac{-3*0cos^{2}(x)}{ln^{4}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)} - \frac{-2(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos^{2}(x)}{ln^{3}{10}lg^{3}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin^{2}(x)} - \frac{-2cos(x)cos^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} - \frac{-2cos(x)sin(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)} - \frac{-2*0cos^{2}(x)}{ln^{3}{10}lg(\frac{sin(x)}{sinh(x)})sin^{2}(x)} - \frac{-(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos^{2}(x)}{ln^{2}{10}lg^{2}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin^{2}(x)} - \frac{-2cos(x)cos^{2}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{3}(x)} - \frac{-2cos(x)sin(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{2}(x)} + \frac{-2*0cosh^{2}(x)}{ln^{3}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} + \frac{-(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cosh^{2}(x)}{ln^{2}{10}lg^{2}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} + \frac{-2cosh(x)cosh^{2}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} + \frac{2cosh(x)sinh(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} - \frac{-3*0cosh^{2}(x)}{ln^{4}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} - \frac{-2(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cosh^{2}(x)}{ln^{3}{10}lg^{3}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} - \frac{-2cosh(x)cosh^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} - \frac{2cosh(x)sinh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} - \frac{2*-2*0}{ln^{3}{10}lg(\frac{sin(x)}{sinh(x)})} - \frac{2*-(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})}{ln^{2}{10}lg^{2}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})}\\=&\frac{-6cos^{2}(x)cosh(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} + \frac{6cos(x)cosh^{2}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} - \frac{3cos^{2}(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} - \frac{6cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh(x)} - \frac{3cos(x)cosh^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} + \frac{6cos(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)} + \frac{3cos^{3}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2cos^{3}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2cos^{3}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2cos(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin(x)} - \frac{2cosh^{3}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} + \frac{3cosh^{3}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} + \frac{2cosh(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh(x)} - \frac{2cosh^{3}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-6cos^{2}(x)cosh(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} + \frac{6cos(x)cosh^{2}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} - \frac{3cos^{2}(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} - \frac{6cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh(x)} - \frac{3cos(x)cosh^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} + \frac{6cos(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)} + \frac{3cos^{3}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2cos^{3}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2cos^{3}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2cos(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin(x)} - \frac{2cosh^{3}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} + \frac{3cosh^{3}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} + \frac{2cosh(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh(x)} - \frac{2cosh^{3}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)}\right)}{dx}\\=&\frac{-6*-4*0cos^{2}(x)cosh(x)}{ln^{5}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} - \frac{6*-3(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos^{2}(x)cosh(x)}{ln^{4}{10}lg^{4}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} - \frac{6*-2cos(x)cos^{2}(x)cosh(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{3}(x)sinh(x)} - \frac{6*-2cos(x)sin(x)cosh(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} - \frac{6cos^{2}(x)*-cosh(x)cosh(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh^{2}(x)} - \frac{6cos^{2}(x)sinh(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} + \frac{6*-4*0cos(x)cosh^{2}(x)}{ln^{5}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} + \frac{6*-3(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos(x)cosh^{2}(x)}{ln^{4}{10}lg^{4}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} + \frac{6*-cos(x)cos(x)cosh^{2}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh^{2}(x)} + \frac{6*-sin(x)cosh^{2}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} + \frac{6cos(x)*-2cosh(x)cosh^{2}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{3}(x)} + \frac{6cos(x)*2cosh(x)sinh(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} - \frac{3*-3*0cos^{2}(x)cosh(x)}{ln^{4}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} - \frac{3*-2(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos^{2}(x)cosh(x)}{ln^{3}{10}lg^{3}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} - \frac{3*-2cos(x)cos^{2}(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{3}(x)sinh(x)} - \frac{3*-2cos(x)sin(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} - \frac{3cos^{2}(x)*-cosh(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh^{2}(x)} - \frac{3cos^{2}(x)sinh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh(x)} - \frac{6*-3*0cosh(x)}{ln^{4}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh(x)} - \frac{6*-2(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cosh(x)}{ln^{3}{10}lg^{3}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sinh(x)} - \frac{6*-cosh(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} - \frac{6sinh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh(x)} - \frac{3*-3*0cos(x)cosh^{2}(x)}{ln^{4}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} - \frac{3*-2(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos(x)cosh^{2}(x)}{ln^{3}{10}lg^{3}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} - \frac{3*-cos(x)cos(x)cosh^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh^{2}(x)} - \frac{3*-sin(x)cosh^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} - \frac{3cos(x)*-2cosh(x)cosh^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{3}(x)} - \frac{3cos(x)*2cosh(x)sinh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{2}(x)} + \frac{6*-3*0cos(x)}{ln^{4}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)} + \frac{6*-2(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos(x)}{ln^{3}{10}lg^{3}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin(x)} + \frac{6*-cos(x)cos(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)} + \frac{6*-sin(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)} + \frac{3*-3*0cos^{3}(x)}{ln^{4}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{3*-2(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos^{3}(x)}{ln^{3}{10}lg^{3}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{3*-3cos(x)cos^{3}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{4}(x)} + \frac{3*-3cos^{2}(x)sin(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2*-4*0cos^{3}(x)}{ln^{5}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2*-3(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos^{3}(x)}{ln^{4}{10}lg^{4}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2*-3cos(x)cos^{3}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{4}(x)} + \frac{2*-3cos^{2}(x)sin(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2*-2*0cos^{3}(x)}{ln^{3}{10}lg(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2*-(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos^{3}(x)}{ln^{2}{10}lg^{2}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2*-3cos(x)cos^{3}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{4}(x)} + \frac{2*-3cos^{2}(x)sin(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{3}(x)} + \frac{2*-2*0cos(x)}{ln^{3}{10}lg(\frac{sin(x)}{sinh(x)})sin(x)} + \frac{2*-(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cos(x)}{ln^{2}{10}lg^{2}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sin(x)} + \frac{2*-cos(x)cos(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{2}(x)} + \frac{2*-sin(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin(x)} - \frac{2*-2*0cosh^{3}(x)}{ln^{3}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} - \frac{2*-(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cosh^{3}(x)}{ln^{2}{10}lg^{2}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} - \frac{2*-3cosh(x)cosh^{3}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{4}(x)} - \frac{2*3cosh^{2}(x)sinh(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} + \frac{3*-3*0cosh^{3}(x)}{ln^{4}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} + \frac{3*-2(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cosh^{3}(x)}{ln^{3}{10}lg^{3}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} + \frac{3*-3cosh(x)cosh^{3}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{4}(x)} + \frac{3*3cosh^{2}(x)sinh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} + \frac{2*-2*0cosh(x)}{ln^{3}{10}lg(\frac{sin(x)}{sinh(x)})sinh(x)} + \frac{2*-(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cosh(x)}{ln^{2}{10}lg^{2}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sinh(x)} + \frac{2*-cosh(x)cosh(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} + \frac{2sinh(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh(x)} - \frac{2*-4*0cosh^{3}(x)}{ln^{5}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} - \frac{2*-3(\frac{cos(x)}{sinh(x)} + \frac{sin(x)*-cosh(x)}{sinh^{2}(x)})cosh^{3}(x)}{ln^{4}{10}lg^{4}(\frac{sin(x)}{sinh(x)})ln{10}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)} - \frac{2*-3cosh(x)cosh^{3}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sinh^{4}(x)} - \frac{2*3cosh^{2}(x)sinh(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sinh^{3}(x)}\\=&\frac{24cos^{3}(x)cosh(x)}{ln^{5}{10}lg^{4}(\frac{sin(x)}{sinh(x)})sin^{3}(x)sinh(x)} - \frac{36cos^{2}(x)cosh^{2}(x)}{ln^{5}{10}lg^{4}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh^{2}(x)} + \frac{24cos^{3}(x)cosh(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{3}(x)sinh(x)} + \frac{48cos(x)cosh(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin(x)sinh(x)} - \frac{24cos(x)cosh^{3}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{3}(x)} + \frac{24cos(x)cosh^{3}(x)}{ln^{5}{10}lg^{4}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{3}(x)} - \frac{24cosh^{2}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} + \frac{6cos^{2}(x)cosh^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)sinh^{2}(x)} + \frac{8cos(x)cosh^{3}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin(x)sinh^{3}(x)} + \frac{8cos^{3}(x)cosh(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{3}(x)sinh(x)} - \frac{20cos^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{2}(x)} + \frac{20cosh^{2}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} + \frac{6cosh^{4}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{4}(x)} - \frac{8cosh^{2}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sinh^{2}(x)} - \frac{24cos^{2}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{2}(x)} - \frac{11cos^{4}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sin^{4}(x)} - \frac{12cos^{4}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sin^{4}(x)} - \frac{6cos^{4}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{4}(x)} - \frac{8cos^{2}(x)}{ln^{2}{10}lg(\frac{sin(x)}{sinh(x)})sin^{2}(x)} - \frac{6cos^{4}(x)}{ln^{5}{10}lg^{4}(\frac{sin(x)}{sinh(x)})sin^{4}(x)} - \frac{11cosh^{4}(x)}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})sinh^{4}(x)} + \frac{12cosh^{4}(x)}{ln^{4}{10}lg^{3}(\frac{sin(x)}{sinh(x)})sinh^{4}(x)} - \frac{6cosh^{4}(x)}{ln^{5}{10}lg^{4}(\frac{sin(x)}{sinh(x)})sinh^{4}(x)} - \frac{12}{ln^{3}{10}lg^{2}(\frac{sin(x)}{sinh(x)})}\\ \end{split}\end{equation} \]