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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ sh(ch(th(x)))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sh(ch(th(x)))\right)}{dx}\\=&ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))\\=& - sh(th(x))ch(ch(th(x)))th^{2}(x) + sh(th(x))ch(ch(th(x)))\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( - sh(th(x))ch(ch(th(x)))th^{2}(x) + sh(th(x))ch(ch(th(x)))\right)}{dx}\\=& - ch(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{2}(x) - sh(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{2}(x) - sh(th(x))ch(ch(th(x)))*2th(x)(1 - th^{2}(x)) + ch(th(x))(1 - th^{2}(x))ch(ch(th(x))) + sh(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))\\=& - 2ch(th(x))ch(ch(th(x)))th^{2}(x) + ch(th(x))ch(ch(th(x)))th^{4}(x) - sh(ch(th(x)))sh^{2}(th(x))th^{2}(x) + sh^{2}(th(x))sh(ch(th(x)))th^{4}(x) - 2sh(th(x))ch(ch(th(x)))th(x) + 2sh(th(x))ch(ch(th(x)))th^{3}(x) + ch(th(x))ch(ch(th(x))) - sh^{2}(th(x))sh(ch(th(x)))th^{2}(x) + sh(ch(th(x)))sh^{2}(th(x))\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( - 2ch(th(x))ch(ch(th(x)))th^{2}(x) + ch(th(x))ch(ch(th(x)))th^{4}(x) - sh(ch(th(x)))sh^{2}(th(x))th^{2}(x) + sh^{2}(th(x))sh(ch(th(x)))th^{4}(x) - 2sh(th(x))ch(ch(th(x)))th(x) + 2sh(th(x))ch(ch(th(x)))th^{3}(x) + ch(th(x))ch(ch(th(x))) - sh^{2}(th(x))sh(ch(th(x)))th^{2}(x) + sh(ch(th(x)))sh^{2}(th(x))\right)}{dx}\\=& - 2sh(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{2}(x) - 2ch(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{2}(x) - 2ch(th(x))ch(ch(th(x)))*2th(x)(1 - th^{2}(x)) + sh(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{4}(x) + ch(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{4}(x) + ch(th(x))ch(ch(th(x)))*4th^{3}(x)(1 - th^{2}(x)) - ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))sh^{2}(th(x))th^{2}(x) - sh(ch(th(x)))*2sh(th(x))ch(th(x))(1 - th^{2}(x))th^{2}(x) - sh(ch(th(x)))sh^{2}(th(x))*2th(x)(1 - th^{2}(x)) + 2sh(th(x))ch(th(x))(1 - th^{2}(x))sh(ch(th(x)))th^{4}(x) + sh^{2}(th(x))ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{4}(x) + sh^{2}(th(x))sh(ch(th(x)))*4th^{3}(x)(1 - th^{2}(x)) - 2ch(th(x))(1 - th^{2}(x))ch(ch(th(x)))th(x) - 2sh(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th(x) - 2sh(th(x))ch(ch(th(x)))(1 - th^{2}(x)) + 2ch(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{3}(x) + 2sh(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{3}(x) + 2sh(th(x))ch(ch(th(x)))*3th^{2}(x)(1 - th^{2}(x)) + sh(th(x))(1 - th^{2}(x))ch(ch(th(x))) + ch(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x)) - 2sh(th(x))ch(th(x))(1 - th^{2}(x))sh(ch(th(x)))th^{2}(x) - sh^{2}(th(x))ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{2}(x) - sh^{2}(th(x))sh(ch(th(x)))*2th(x)(1 - th^{2}(x)) + ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))sh^{2}(th(x)) + sh(ch(th(x)))*2sh(th(x))ch(th(x))(1 - th^{2}(x))\\=&5sh(th(x))ch(ch(th(x)))th^{2}(x) - 3sh(th(x))ch(ch(th(x)))th^{4}(x) - 4sh(ch(th(x)))sh(th(x))ch(th(x))th^{2}(x) + 8sh(th(x))sh(ch(th(x)))ch(th(x))th^{4}(x) - 4ch(ch(th(x)))ch(th(x))th(x) + 8ch(th(x))ch(ch(th(x)))th^{3}(x) - sh(th(x))ch(ch(th(x)))th^{6}(x) + sh(ch(th(x)))sh(th(x))ch(th(x))th^{4}(x) - 3sh(th(x))sh(ch(th(x)))ch(th(x))th^{6}(x) + 4ch(ch(th(x)))ch(th(x))th^{3}(x) - 6ch(th(x))ch(ch(th(x)))th^{5}(x) - 3sh^{3}(th(x))ch(ch(th(x)))th^{2}(x) + 3sh^{3}(th(x))ch(ch(th(x)))th^{4}(x) - 2sh^{2}(th(x))sh(ch(th(x)))th(x) + 8sh(ch(th(x)))sh^{2}(th(x))th^{3}(x) - sh^{3}(th(x))ch(ch(th(x)))th^{6}(x) - 6sh^{2}(th(x))sh(ch(th(x)))th^{5}(x) - 2ch(th(x))ch(ch(th(x)))th(x) - 4sh(ch(th(x)))sh^{2}(th(x))th(x) + 4sh^{2}(th(x))sh(ch(th(x)))th^{3}(x) - sh(th(x))ch(ch(th(x))) - 5sh(th(x))sh(ch(th(x)))ch(th(x))th^{2}(x) + 3sh(ch(th(x)))sh(th(x))ch(th(x)) + sh^{3}(th(x))ch(ch(th(x)))\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 5sh(th(x))ch(ch(th(x)))th^{2}(x) - 3sh(th(x))ch(ch(th(x)))th^{4}(x) - 4sh(ch(th(x)))sh(th(x))ch(th(x))th^{2}(x) + 8sh(th(x))sh(ch(th(x)))ch(th(x))th^{4}(x) - 4ch(ch(th(x)))ch(th(x))th(x) + 8ch(th(x))ch(ch(th(x)))th^{3}(x) - sh(th(x))ch(ch(th(x)))th^{6}(x) + sh(ch(th(x)))sh(th(x))ch(th(x))th^{4}(x) - 3sh(th(x))sh(ch(th(x)))ch(th(x))th^{6}(x) + 4ch(ch(th(x)))ch(th(x))th^{3}(x) - 6ch(th(x))ch(ch(th(x)))th^{5}(x) - 3sh^{3}(th(x))ch(ch(th(x)))th^{2}(x) + 3sh^{3}(th(x))ch(ch(th(x)))th^{4}(x) - 2sh^{2}(th(x))sh(ch(th(x)))th(x) + 8sh(ch(th(x)))sh^{2}(th(x))th^{3}(x) - sh^{3}(th(x))ch(ch(th(x)))th^{6}(x) - 6sh^{2}(th(x))sh(ch(th(x)))th^{5}(x) - 2ch(th(x))ch(ch(th(x)))th(x) - 4sh(ch(th(x)))sh^{2}(th(x))th(x) + 4sh^{2}(th(x))sh(ch(th(x)))th^{3}(x) - sh(th(x))ch(ch(th(x))) - 5sh(th(x))sh(ch(th(x)))ch(th(x))th^{2}(x) + 3sh(ch(th(x)))sh(th(x))ch(th(x)) + sh^{3}(th(x))ch(ch(th(x)))\right)}{dx}\\=&5ch(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{2}(x) + 5sh(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{2}(x) + 5sh(th(x))ch(ch(th(x)))*2th(x)(1 - th^{2}(x)) - 3ch(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{4}(x) - 3sh(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{4}(x) - 3sh(th(x))ch(ch(th(x)))*4th^{3}(x)(1 - th^{2}(x)) - 4ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))sh(th(x))ch(th(x))th^{2}(x) - 4sh(ch(th(x)))ch(th(x))(1 - th^{2}(x))ch(th(x))th^{2}(x) - 4sh(ch(th(x)))sh(th(x))sh(th(x))(1 - th^{2}(x))th^{2}(x) - 4sh(ch(th(x)))sh(th(x))ch(th(x))*2th(x)(1 - th^{2}(x)) + 8ch(th(x))(1 - th^{2}(x))sh(ch(th(x)))ch(th(x))th^{4}(x) + 8sh(th(x))ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))ch(th(x))th^{4}(x) + 8sh(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{4}(x) + 8sh(th(x))sh(ch(th(x)))ch(th(x))*4th^{3}(x)(1 - th^{2}(x)) - 4sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))ch(th(x))th(x) - 4ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))th(x) - 4ch(ch(th(x)))ch(th(x))(1 - th^{2}(x)) + 8sh(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{3}(x) + 8ch(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{3}(x) + 8ch(th(x))ch(ch(th(x)))*3th^{2}(x)(1 - th^{2}(x)) - ch(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{6}(x) - sh(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{6}(x) - sh(th(x))ch(ch(th(x)))*6th^{5}(x)(1 - th^{2}(x)) + ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))sh(th(x))ch(th(x))th^{4}(x) + sh(ch(th(x)))ch(th(x))(1 - th^{2}(x))ch(th(x))th^{4}(x) + sh(ch(th(x)))sh(th(x))sh(th(x))(1 - th^{2}(x))th^{4}(x) + sh(ch(th(x)))sh(th(x))ch(th(x))*4th^{3}(x)(1 - th^{2}(x)) - 3ch(th(x))(1 - th^{2}(x))sh(ch(th(x)))ch(th(x))th^{6}(x) - 3sh(th(x))ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))ch(th(x))th^{6}(x) - 3sh(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{6}(x) - 3sh(th(x))sh(ch(th(x)))ch(th(x))*6th^{5}(x)(1 - th^{2}(x)) + 4sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))ch(th(x))th^{3}(x) + 4ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{3}(x) + 4ch(ch(th(x)))ch(th(x))*3th^{2}(x)(1 - th^{2}(x)) - 6sh(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{5}(x) - 6ch(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{5}(x) - 6ch(th(x))ch(ch(th(x)))*5th^{4}(x)(1 - th^{2}(x)) - 3*3sh^{2}(th(x))ch(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{2}(x) - 3sh^{3}(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{2}(x) - 3sh^{3}(th(x))ch(ch(th(x)))*2th(x)(1 - th^{2}(x)) + 3*3sh^{2}(th(x))ch(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{4}(x) + 3sh^{3}(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{4}(x) + 3sh^{3}(th(x))ch(ch(th(x)))*4th^{3}(x)(1 - th^{2}(x)) - 2*2sh(th(x))ch(th(x))(1 - th^{2}(x))sh(ch(th(x)))th(x) - 2sh^{2}(th(x))ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))th(x) - 2sh^{2}(th(x))sh(ch(th(x)))(1 - th^{2}(x)) + 8ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))sh^{2}(th(x))th^{3}(x) + 8sh(ch(th(x)))*2sh(th(x))ch(th(x))(1 - th^{2}(x))th^{3}(x) + 8sh(ch(th(x)))sh^{2}(th(x))*3th^{2}(x)(1 - th^{2}(x)) - 3sh^{2}(th(x))ch(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{6}(x) - sh^{3}(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{6}(x) - sh^{3}(th(x))ch(ch(th(x)))*6th^{5}(x)(1 - th^{2}(x)) - 6*2sh(th(x))ch(th(x))(1 - th^{2}(x))sh(ch(th(x)))th^{5}(x) - 6sh^{2}(th(x))ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{5}(x) - 6sh^{2}(th(x))sh(ch(th(x)))*5th^{4}(x)(1 - th^{2}(x)) - 2sh(th(x))(1 - th^{2}(x))ch(ch(th(x)))th(x) - 2ch(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th(x) - 2ch(th(x))ch(ch(th(x)))(1 - th^{2}(x)) - 4ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))sh^{2}(th(x))th(x) - 4sh(ch(th(x)))*2sh(th(x))ch(th(x))(1 - th^{2}(x))th(x) - 4sh(ch(th(x)))sh^{2}(th(x))(1 - th^{2}(x)) + 4*2sh(th(x))ch(th(x))(1 - th^{2}(x))sh(ch(th(x)))th^{3}(x) + 4sh^{2}(th(x))ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{3}(x) + 4sh^{2}(th(x))sh(ch(th(x)))*3th^{2}(x)(1 - th^{2}(x)) - ch(th(x))(1 - th^{2}(x))ch(ch(th(x))) - sh(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x)) - 5ch(th(x))(1 - th^{2}(x))sh(ch(th(x)))ch(th(x))th^{2}(x) - 5sh(th(x))ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))ch(th(x))th^{2}(x) - 5sh(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{2}(x) - 5sh(th(x))sh(ch(th(x)))ch(th(x))*2th(x)(1 - th^{2}(x)) + 3ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))sh(th(x))ch(th(x)) + 3sh(ch(th(x)))ch(th(x))(1 - th^{2}(x))ch(th(x)) + 3sh(ch(th(x)))sh(th(x))sh(th(x))(1 - th^{2}(x)) + 3sh^{2}(th(x))ch(th(x))(1 - th^{2}(x))ch(ch(th(x))) + sh^{3}(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))\\=&20ch(th(x))ch(ch(th(x)))th^{2}(x) - 32ch(th(x))ch(ch(th(x)))th^{4}(x) + 17sh(ch(th(x)))sh^{2}(th(x))th^{2}(x) - 5sh^{2}(th(x))sh(ch(th(x)))th^{4}(x) + 4sh(th(x))ch(ch(th(x)))th(x) - 4sh(th(x))ch(ch(th(x)))th^{3}(x) + 32ch(th(x))ch(ch(th(x)))th^{6}(x) - 51sh(ch(th(x)))sh^{2}(th(x))th^{4}(x) + 29sh^{2}(th(x))sh(ch(th(x)))th^{6}(x) - 12sh(th(x))ch(ch(th(x)))th^{5}(x) - 12sh^{2}(th(x))ch(ch(th(x)))ch(th(x))th^{2}(x) + 18sh^{2}(th(x))ch(ch(th(x)))ch(th(x))th^{4}(x) - 12sh(ch(th(x)))ch^{2}(th(x))th^{2}(x) + 18sh(ch(th(x)))ch^{2}(th(x))th^{4}(x) - 12sh(th(x))sh(ch(th(x)))ch(th(x))th(x) + 68sh(ch(th(x)))sh(th(x))ch(th(x))th^{3}(x) - 12sh(ch(th(x)))ch^{2}(th(x))th^{6}(x) - 12sh^{2}(th(x))ch(ch(th(x)))ch(th(x))th^{6}(x) - 9sh(ch(th(x)))sh^{2}(th(x))th^{6}(x) - 80sh(th(x))sh(ch(th(x)))ch(th(x))th^{5}(x) - 24sh(ch(th(x)))sh(th(x))ch(th(x))th(x) + 40sh(th(x))sh(ch(th(x)))ch(th(x))th^{3}(x) + 28ch(ch(th(x)))ch(th(x))th^{2}(x) - 42ch(ch(th(x)))ch(th(x))th^{4}(x) + ch(th(x))ch(ch(th(x)))th^{8}(x) + sh^{2}(th(x))sh(ch(th(x)))th^{8}(x) + 12sh(th(x))ch(ch(th(x)))th^{7}(x) - 28sh(ch(th(x)))sh(th(x))ch(th(x))th^{5}(x) + 3sh(ch(th(x)))ch^{2}(th(x))th^{8}(x) + 3sh^{2}(th(x))ch(ch(th(x)))ch(th(x))th^{8}(x) + 3sh(ch(th(x)))sh^{2}(th(x))th^{8}(x) + 36sh(th(x))sh(ch(th(x)))ch(th(x))th^{7}(x) - 5ch(th(x))ch(ch(th(x))) - 12sh^{2}(th(x))ch(th(x))ch(ch(th(x)))th^{2}(x) + 18sh^{2}(th(x))ch(th(x))ch(ch(th(x)))th^{4}(x) - 3sh(ch(th(x)))sh^{4}(th(x))th^{2}(x) + 3sh^{4}(th(x))sh(ch(th(x)))th^{4}(x) - 12sh^{3}(th(x))ch(ch(th(x)))th(x) + 36sh^{3}(th(x))ch(ch(th(x)))th^{3}(x) - 12sh^{2}(th(x))ch(th(x))ch(ch(th(x)))th^{6}(x) + 3sh(ch(th(x)))sh^{4}(th(x))th^{4}(x) - 3sh^{4}(th(x))sh(ch(th(x)))th^{6}(x) - 36sh^{3}(th(x))ch(ch(th(x)))th^{5}(x) + 19sh^{2}(th(x))sh(ch(th(x)))th^{2}(x) + 3sh^{2}(th(x))ch(th(x))ch(ch(th(x)))th^{8}(x) - sh(ch(th(x)))sh^{4}(th(x))th^{6}(x) + sh^{4}(th(x))sh(ch(th(x)))th^{8}(x) + 12sh^{3}(th(x))ch(ch(th(x)))th^{7}(x) - 2ch(ch(th(x)))ch(th(x)) - sh^{4}(th(x))sh(ch(th(x)))th^{2}(x) + 3sh^{2}(th(x))ch(ch(th(x)))ch(th(x)) + 3sh^{2}(th(x))ch(th(x))ch(ch(th(x))) + 3sh(ch(th(x)))ch^{2}(th(x)) + sh(ch(th(x)))sh^{4}(th(x)) - 4sh^{2}(th(x))sh(ch(th(x)))\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ sh(ch(th(x)))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sh(ch(th(x)))\right)}{dx}\\=&ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))\\=& - sh(th(x))ch(ch(th(x)))th^{2}(x) + sh(th(x))ch(ch(th(x)))\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( - sh(th(x))ch(ch(th(x)))th^{2}(x) + sh(th(x))ch(ch(th(x)))\right)}{dx}\\=& - ch(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{2}(x) - sh(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{2}(x) - sh(th(x))ch(ch(th(x)))*2th(x)(1 - th^{2}(x)) + ch(th(x))(1 - th^{2}(x))ch(ch(th(x))) + sh(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))\\=& - 2ch(th(x))ch(ch(th(x)))th^{2}(x) + ch(th(x))ch(ch(th(x)))th^{4}(x) - sh(ch(th(x)))sh^{2}(th(x))th^{2}(x) + sh^{2}(th(x))sh(ch(th(x)))th^{4}(x) - 2sh(th(x))ch(ch(th(x)))th(x) + 2sh(th(x))ch(ch(th(x)))th^{3}(x) + ch(th(x))ch(ch(th(x))) - sh^{2}(th(x))sh(ch(th(x)))th^{2}(x) + sh(ch(th(x)))sh^{2}(th(x))\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( - 2ch(th(x))ch(ch(th(x)))th^{2}(x) + ch(th(x))ch(ch(th(x)))th^{4}(x) - sh(ch(th(x)))sh^{2}(th(x))th^{2}(x) + sh^{2}(th(x))sh(ch(th(x)))th^{4}(x) - 2sh(th(x))ch(ch(th(x)))th(x) + 2sh(th(x))ch(ch(th(x)))th^{3}(x) + ch(th(x))ch(ch(th(x))) - sh^{2}(th(x))sh(ch(th(x)))th^{2}(x) + sh(ch(th(x)))sh^{2}(th(x))\right)}{dx}\\=& - 2sh(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{2}(x) - 2ch(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{2}(x) - 2ch(th(x))ch(ch(th(x)))*2th(x)(1 - th^{2}(x)) + sh(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{4}(x) + ch(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{4}(x) + ch(th(x))ch(ch(th(x)))*4th^{3}(x)(1 - th^{2}(x)) - ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))sh^{2}(th(x))th^{2}(x) - sh(ch(th(x)))*2sh(th(x))ch(th(x))(1 - th^{2}(x))th^{2}(x) - sh(ch(th(x)))sh^{2}(th(x))*2th(x)(1 - th^{2}(x)) + 2sh(th(x))ch(th(x))(1 - th^{2}(x))sh(ch(th(x)))th^{4}(x) + sh^{2}(th(x))ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{4}(x) + sh^{2}(th(x))sh(ch(th(x)))*4th^{3}(x)(1 - th^{2}(x)) - 2ch(th(x))(1 - th^{2}(x))ch(ch(th(x)))th(x) - 2sh(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th(x) - 2sh(th(x))ch(ch(th(x)))(1 - th^{2}(x)) + 2ch(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{3}(x) + 2sh(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{3}(x) + 2sh(th(x))ch(ch(th(x)))*3th^{2}(x)(1 - th^{2}(x)) + sh(th(x))(1 - th^{2}(x))ch(ch(th(x))) + ch(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x)) - 2sh(th(x))ch(th(x))(1 - th^{2}(x))sh(ch(th(x)))th^{2}(x) - sh^{2}(th(x))ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{2}(x) - sh^{2}(th(x))sh(ch(th(x)))*2th(x)(1 - th^{2}(x)) + ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))sh^{2}(th(x)) + sh(ch(th(x)))*2sh(th(x))ch(th(x))(1 - th^{2}(x))\\=&5sh(th(x))ch(ch(th(x)))th^{2}(x) - 3sh(th(x))ch(ch(th(x)))th^{4}(x) - 4sh(ch(th(x)))sh(th(x))ch(th(x))th^{2}(x) + 8sh(th(x))sh(ch(th(x)))ch(th(x))th^{4}(x) - 4ch(ch(th(x)))ch(th(x))th(x) + 8ch(th(x))ch(ch(th(x)))th^{3}(x) - sh(th(x))ch(ch(th(x)))th^{6}(x) + sh(ch(th(x)))sh(th(x))ch(th(x))th^{4}(x) - 3sh(th(x))sh(ch(th(x)))ch(th(x))th^{6}(x) + 4ch(ch(th(x)))ch(th(x))th^{3}(x) - 6ch(th(x))ch(ch(th(x)))th^{5}(x) - 3sh^{3}(th(x))ch(ch(th(x)))th^{2}(x) + 3sh^{3}(th(x))ch(ch(th(x)))th^{4}(x) - 2sh^{2}(th(x))sh(ch(th(x)))th(x) + 8sh(ch(th(x)))sh^{2}(th(x))th^{3}(x) - sh^{3}(th(x))ch(ch(th(x)))th^{6}(x) - 6sh^{2}(th(x))sh(ch(th(x)))th^{5}(x) - 2ch(th(x))ch(ch(th(x)))th(x) - 4sh(ch(th(x)))sh^{2}(th(x))th(x) + 4sh^{2}(th(x))sh(ch(th(x)))th^{3}(x) - sh(th(x))ch(ch(th(x))) - 5sh(th(x))sh(ch(th(x)))ch(th(x))th^{2}(x) + 3sh(ch(th(x)))sh(th(x))ch(th(x)) + sh^{3}(th(x))ch(ch(th(x)))\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 5sh(th(x))ch(ch(th(x)))th^{2}(x) - 3sh(th(x))ch(ch(th(x)))th^{4}(x) - 4sh(ch(th(x)))sh(th(x))ch(th(x))th^{2}(x) + 8sh(th(x))sh(ch(th(x)))ch(th(x))th^{4}(x) - 4ch(ch(th(x)))ch(th(x))th(x) + 8ch(th(x))ch(ch(th(x)))th^{3}(x) - sh(th(x))ch(ch(th(x)))th^{6}(x) + sh(ch(th(x)))sh(th(x))ch(th(x))th^{4}(x) - 3sh(th(x))sh(ch(th(x)))ch(th(x))th^{6}(x) + 4ch(ch(th(x)))ch(th(x))th^{3}(x) - 6ch(th(x))ch(ch(th(x)))th^{5}(x) - 3sh^{3}(th(x))ch(ch(th(x)))th^{2}(x) + 3sh^{3}(th(x))ch(ch(th(x)))th^{4}(x) - 2sh^{2}(th(x))sh(ch(th(x)))th(x) + 8sh(ch(th(x)))sh^{2}(th(x))th^{3}(x) - sh^{3}(th(x))ch(ch(th(x)))th^{6}(x) - 6sh^{2}(th(x))sh(ch(th(x)))th^{5}(x) - 2ch(th(x))ch(ch(th(x)))th(x) - 4sh(ch(th(x)))sh^{2}(th(x))th(x) + 4sh^{2}(th(x))sh(ch(th(x)))th^{3}(x) - sh(th(x))ch(ch(th(x))) - 5sh(th(x))sh(ch(th(x)))ch(th(x))th^{2}(x) + 3sh(ch(th(x)))sh(th(x))ch(th(x)) + sh^{3}(th(x))ch(ch(th(x)))\right)}{dx}\\=&5ch(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{2}(x) + 5sh(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{2}(x) + 5sh(th(x))ch(ch(th(x)))*2th(x)(1 - th^{2}(x)) - 3ch(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{4}(x) - 3sh(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{4}(x) - 3sh(th(x))ch(ch(th(x)))*4th^{3}(x)(1 - th^{2}(x)) - 4ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))sh(th(x))ch(th(x))th^{2}(x) - 4sh(ch(th(x)))ch(th(x))(1 - th^{2}(x))ch(th(x))th^{2}(x) - 4sh(ch(th(x)))sh(th(x))sh(th(x))(1 - th^{2}(x))th^{2}(x) - 4sh(ch(th(x)))sh(th(x))ch(th(x))*2th(x)(1 - th^{2}(x)) + 8ch(th(x))(1 - th^{2}(x))sh(ch(th(x)))ch(th(x))th^{4}(x) + 8sh(th(x))ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))ch(th(x))th^{4}(x) + 8sh(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{4}(x) + 8sh(th(x))sh(ch(th(x)))ch(th(x))*4th^{3}(x)(1 - th^{2}(x)) - 4sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))ch(th(x))th(x) - 4ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))th(x) - 4ch(ch(th(x)))ch(th(x))(1 - th^{2}(x)) + 8sh(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{3}(x) + 8ch(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{3}(x) + 8ch(th(x))ch(ch(th(x)))*3th^{2}(x)(1 - th^{2}(x)) - ch(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{6}(x) - sh(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{6}(x) - sh(th(x))ch(ch(th(x)))*6th^{5}(x)(1 - th^{2}(x)) + ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))sh(th(x))ch(th(x))th^{4}(x) + sh(ch(th(x)))ch(th(x))(1 - th^{2}(x))ch(th(x))th^{4}(x) + sh(ch(th(x)))sh(th(x))sh(th(x))(1 - th^{2}(x))th^{4}(x) + sh(ch(th(x)))sh(th(x))ch(th(x))*4th^{3}(x)(1 - th^{2}(x)) - 3ch(th(x))(1 - th^{2}(x))sh(ch(th(x)))ch(th(x))th^{6}(x) - 3sh(th(x))ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))ch(th(x))th^{6}(x) - 3sh(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{6}(x) - 3sh(th(x))sh(ch(th(x)))ch(th(x))*6th^{5}(x)(1 - th^{2}(x)) + 4sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))ch(th(x))th^{3}(x) + 4ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{3}(x) + 4ch(ch(th(x)))ch(th(x))*3th^{2}(x)(1 - th^{2}(x)) - 6sh(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{5}(x) - 6ch(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{5}(x) - 6ch(th(x))ch(ch(th(x)))*5th^{4}(x)(1 - th^{2}(x)) - 3*3sh^{2}(th(x))ch(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{2}(x) - 3sh^{3}(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{2}(x) - 3sh^{3}(th(x))ch(ch(th(x)))*2th(x)(1 - th^{2}(x)) + 3*3sh^{2}(th(x))ch(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{4}(x) + 3sh^{3}(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{4}(x) + 3sh^{3}(th(x))ch(ch(th(x)))*4th^{3}(x)(1 - th^{2}(x)) - 2*2sh(th(x))ch(th(x))(1 - th^{2}(x))sh(ch(th(x)))th(x) - 2sh^{2}(th(x))ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))th(x) - 2sh^{2}(th(x))sh(ch(th(x)))(1 - th^{2}(x)) + 8ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))sh^{2}(th(x))th^{3}(x) + 8sh(ch(th(x)))*2sh(th(x))ch(th(x))(1 - th^{2}(x))th^{3}(x) + 8sh(ch(th(x)))sh^{2}(th(x))*3th^{2}(x)(1 - th^{2}(x)) - 3sh^{2}(th(x))ch(th(x))(1 - th^{2}(x))ch(ch(th(x)))th^{6}(x) - sh^{3}(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{6}(x) - sh^{3}(th(x))ch(ch(th(x)))*6th^{5}(x)(1 - th^{2}(x)) - 6*2sh(th(x))ch(th(x))(1 - th^{2}(x))sh(ch(th(x)))th^{5}(x) - 6sh^{2}(th(x))ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{5}(x) - 6sh^{2}(th(x))sh(ch(th(x)))*5th^{4}(x)(1 - th^{2}(x)) - 2sh(th(x))(1 - th^{2}(x))ch(ch(th(x)))th(x) - 2ch(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th(x) - 2ch(th(x))ch(ch(th(x)))(1 - th^{2}(x)) - 4ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))sh^{2}(th(x))th(x) - 4sh(ch(th(x)))*2sh(th(x))ch(th(x))(1 - th^{2}(x))th(x) - 4sh(ch(th(x)))sh^{2}(th(x))(1 - th^{2}(x)) + 4*2sh(th(x))ch(th(x))(1 - th^{2}(x))sh(ch(th(x)))th^{3}(x) + 4sh^{2}(th(x))ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{3}(x) + 4sh^{2}(th(x))sh(ch(th(x)))*3th^{2}(x)(1 - th^{2}(x)) - ch(th(x))(1 - th^{2}(x))ch(ch(th(x))) - sh(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x)) - 5ch(th(x))(1 - th^{2}(x))sh(ch(th(x)))ch(th(x))th^{2}(x) - 5sh(th(x))ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))ch(th(x))th^{2}(x) - 5sh(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))th^{2}(x) - 5sh(th(x))sh(ch(th(x)))ch(th(x))*2th(x)(1 - th^{2}(x)) + 3ch(ch(th(x)))sh(th(x))(1 - th^{2}(x))sh(th(x))ch(th(x)) + 3sh(ch(th(x)))ch(th(x))(1 - th^{2}(x))ch(th(x)) + 3sh(ch(th(x)))sh(th(x))sh(th(x))(1 - th^{2}(x)) + 3sh^{2}(th(x))ch(th(x))(1 - th^{2}(x))ch(ch(th(x))) + sh^{3}(th(x))sh(ch(th(x)))sh(th(x))(1 - th^{2}(x))\\=&20ch(th(x))ch(ch(th(x)))th^{2}(x) - 32ch(th(x))ch(ch(th(x)))th^{4}(x) + 17sh(ch(th(x)))sh^{2}(th(x))th^{2}(x) - 5sh^{2}(th(x))sh(ch(th(x)))th^{4}(x) + 4sh(th(x))ch(ch(th(x)))th(x) - 4sh(th(x))ch(ch(th(x)))th^{3}(x) + 32ch(th(x))ch(ch(th(x)))th^{6}(x) - 51sh(ch(th(x)))sh^{2}(th(x))th^{4}(x) + 29sh^{2}(th(x))sh(ch(th(x)))th^{6}(x) - 12sh(th(x))ch(ch(th(x)))th^{5}(x) - 12sh^{2}(th(x))ch(ch(th(x)))ch(th(x))th^{2}(x) + 18sh^{2}(th(x))ch(ch(th(x)))ch(th(x))th^{4}(x) - 12sh(ch(th(x)))ch^{2}(th(x))th^{2}(x) + 18sh(ch(th(x)))ch^{2}(th(x))th^{4}(x) - 12sh(th(x))sh(ch(th(x)))ch(th(x))th(x) + 68sh(ch(th(x)))sh(th(x))ch(th(x))th^{3}(x) - 12sh(ch(th(x)))ch^{2}(th(x))th^{6}(x) - 12sh^{2}(th(x))ch(ch(th(x)))ch(th(x))th^{6}(x) - 9sh(ch(th(x)))sh^{2}(th(x))th^{6}(x) - 80sh(th(x))sh(ch(th(x)))ch(th(x))th^{5}(x) - 24sh(ch(th(x)))sh(th(x))ch(th(x))th(x) + 40sh(th(x))sh(ch(th(x)))ch(th(x))th^{3}(x) + 28ch(ch(th(x)))ch(th(x))th^{2}(x) - 42ch(ch(th(x)))ch(th(x))th^{4}(x) + ch(th(x))ch(ch(th(x)))th^{8}(x) + sh^{2}(th(x))sh(ch(th(x)))th^{8}(x) + 12sh(th(x))ch(ch(th(x)))th^{7}(x) - 28sh(ch(th(x)))sh(th(x))ch(th(x))th^{5}(x) + 3sh(ch(th(x)))ch^{2}(th(x))th^{8}(x) + 3sh^{2}(th(x))ch(ch(th(x)))ch(th(x))th^{8}(x) + 3sh(ch(th(x)))sh^{2}(th(x))th^{8}(x) + 36sh(th(x))sh(ch(th(x)))ch(th(x))th^{7}(x) - 5ch(th(x))ch(ch(th(x))) - 12sh^{2}(th(x))ch(th(x))ch(ch(th(x)))th^{2}(x) + 18sh^{2}(th(x))ch(th(x))ch(ch(th(x)))th^{4}(x) - 3sh(ch(th(x)))sh^{4}(th(x))th^{2}(x) + 3sh^{4}(th(x))sh(ch(th(x)))th^{4}(x) - 12sh^{3}(th(x))ch(ch(th(x)))th(x) + 36sh^{3}(th(x))ch(ch(th(x)))th^{3}(x) - 12sh^{2}(th(x))ch(th(x))ch(ch(th(x)))th^{6}(x) + 3sh(ch(th(x)))sh^{4}(th(x))th^{4}(x) - 3sh^{4}(th(x))sh(ch(th(x)))th^{6}(x) - 36sh^{3}(th(x))ch(ch(th(x)))th^{5}(x) + 19sh^{2}(th(x))sh(ch(th(x)))th^{2}(x) + 3sh^{2}(th(x))ch(th(x))ch(ch(th(x)))th^{8}(x) - sh(ch(th(x)))sh^{4}(th(x))th^{6}(x) + sh^{4}(th(x))sh(ch(th(x)))th^{8}(x) + 12sh^{3}(th(x))ch(ch(th(x)))th^{7}(x) - 2ch(ch(th(x)))ch(th(x)) - sh^{4}(th(x))sh(ch(th(x)))th^{2}(x) + 3sh^{2}(th(x))ch(ch(th(x)))ch(th(x)) + 3sh^{2}(th(x))ch(th(x))ch(ch(th(x))) + 3sh(ch(th(x)))ch^{2}(th(x)) + sh(ch(th(x)))sh^{4}(th(x)) - 4sh^{2}(th(x))sh(ch(th(x)))\\ \end{split}\end{equation} \]