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 6 questions in this calculation: for each question, the 4 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/6]Find\ the\ 4th\ derivative\ of\ function\ sech(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sech(x)\right)}{dx}\\=&-sech(x)tanh(x)\\=&-tanh(x)sech(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -tanh(x)sech(x)\right)}{dx}\\=&-sech^{2}(x)sech(x) - tanh(x)*-sech(x)tanh(x)\\=&-sech^{3}(x) + tanh^{2}(x)sech(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -sech^{3}(x) + tanh^{2}(x)sech(x)\right)}{dx}\\=&--3sech^{2}(x)sech(x)tanh(x) + 2tanh(x)sech^{2}(x)sech(x) + tanh^{2}(x)*-sech(x)tanh(x)\\=&5tanh(x)sech^{3}(x) - tanh^{3}(x)sech(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 5tanh(x)sech^{3}(x) - tanh^{3}(x)sech(x)\right)}{dx}\\=&5sech^{2}(x)sech^{3}(x) + 5tanh(x)*-3sech^{2}(x)sech(x)tanh(x) - 3tanh^{2}(x)sech^{2}(x)sech(x) - tanh^{3}(x)*-sech(x)tanh(x)\\=&5sech^{5}(x) - 18tanh^{2}(x)sech^{3}(x) + tanh^{4}(x)sech(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/6]Find\ the\ 4th\ derivative\ of\ function\ csch(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( csch(x)\right)}{dx}\\=&-csch(x)coth(x)\\=&-coth(x)csch(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -coth(x)csch(x)\right)}{dx}\\=&--csch^{2}(x)csch(x) - coth(x)*-csch(x)coth(x)\\=&csch^{3}(x) + coth^{2}(x)csch(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( csch^{3}(x) + coth^{2}(x)csch(x)\right)}{dx}\\=&-3csch^{2}(x)csch(x)coth(x) + -2coth(x)csch^{2}(x)csch(x) + coth^{2}(x)*-csch(x)coth(x)\\=&-5coth(x)csch^{3}(x) - coth^{3}(x)csch(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -5coth(x)csch^{3}(x) - coth^{3}(x)csch(x)\right)}{dx}\\=&-5*-csch^{2}(x)csch^{3}(x) - 5coth(x)*-3csch^{2}(x)csch(x)coth(x) - -3coth^{2}(x)csch^{2}(x)csch(x) - coth^{3}(x)*-csch(x)coth(x)\\=&5csch^{5}(x) + 18coth^{2}(x)csch^{3}(x) + coth^{4}(x)csch(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/6]Find\ the\ 4th\ derivative\ of\ function\ tanh(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( tanh(x)\right)}{dx}\\=&sech^{2}(x)\\=&sech^{2}(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( sech^{2}(x)\right)}{dx}\\=&-2sech(x)sech(x)tanh(x)\\=&-2tanh(x)sech^{2}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -2tanh(x)sech^{2}(x)\right)}{dx}\\=&-2sech^{2}(x)sech^{2}(x) - 2tanh(x)*-2sech(x)sech(x)tanh(x)\\=&-2sech^{4}(x) + 4tanh^{2}(x)sech^{2}(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -2sech^{4}(x) + 4tanh^{2}(x)sech^{2}(x)\right)}{dx}\\=&-2*-4sech^{3}(x)sech(x)tanh(x) + 4*2tanh(x)sech^{2}(x)sech^{2}(x) + 4tanh^{2}(x)*-2sech(x)sech(x)tanh(x)\\=&16tanh(x)sech^{4}(x) - 8tanh^{3}(x)sech^{2}(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[4/6]Find\ the\ 4th\ derivative\ of\ function\ th(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( th(x)\right)}{dx}\\=&(1 - th^{2}(x))\\=& - th^{2}(x) + 1\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( - th^{2}(x) + 1\right)}{dx}\\=& - 2th(x)(1 - th^{2}(x)) + 0\\=& - 2th(x) + 2th^{3}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( - 2th(x) + 2th^{3}(x)\right)}{dx}\\=& - 2(1 - th^{2}(x)) + 2*3th^{2}(x)(1 - th^{2}(x))\\=&8th^{2}(x) - 6th^{4}(x) - 2\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 8th^{2}(x) - 6th^{4}(x) - 2\right)}{dx}\\=&8*2th(x)(1 - th^{2}(x)) - 6*4th^{3}(x)(1 - th^{2}(x)) + 0\\=&16th(x) - 40th^{3}(x) + 24th^{5}(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[5/6]Find\ the\ 4th\ derivative\ of\ function\ coth(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( coth(x)\right)}{dx}\\=&-csch^{2}(x)\\=&-csch^{2}(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -csch^{2}(x)\right)}{dx}\\=&--2csch(x)csch(x)coth(x)\\=&2coth(x)csch^{2}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2coth(x)csch^{2}(x)\right)}{dx}\\=&2*-csch^{2}(x)csch^{2}(x) + 2coth(x)*-2csch(x)csch(x)coth(x)\\=&-2csch^{4}(x) - 4coth^{2}(x)csch^{2}(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -2csch^{4}(x) - 4coth^{2}(x)csch^{2}(x)\right)}{dx}\\=&-2*-4csch^{3}(x)csch(x)coth(x) - 4*-2coth(x)csch^{2}(x)csch^{2}(x) - 4coth^{2}(x)*-2csch(x)csch(x)coth(x)\\=&16coth(x)csch^{4}(x) + 8coth^{3}(x)csch^{2}(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[6/6]Find\ the\ 4th\ derivative\ of\ function\ cth(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cth(x)\right)}{dx}\\=&(1 - cth^{2}(x))\\=& - cth^{2}(x) + 1\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( - cth^{2}(x) + 1\right)}{dx}\\=& - 2cth(x)(1 - cth^{2}(x)) + 0\\=& - 2cth(x) + 2cth^{3}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( - 2cth(x) + 2cth^{3}(x)\right)}{dx}\\=& - 2(1 - cth^{2}(x)) + 2*3cth^{2}(x)(1 - cth^{2}(x))\\=&8cth^{2}(x) - 6cth^{4}(x) - 2\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 8cth^{2}(x) - 6cth^{4}(x) - 2\right)}{dx}\\=&8*2cth(x)(1 - cth^{2}(x)) - 6*4cth^{3}(x)(1 - cth^{2}(x)) + 0\\=&16cth(x) - 40cth^{3}(x) + 24cth^{5}(x)\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/6]Find\ the\ 4th\ derivative\ of\ function\ sech(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sech(x)\right)}{dx}\\=&-sech(x)tanh(x)\\=&-tanh(x)sech(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -tanh(x)sech(x)\right)}{dx}\\=&-sech^{2}(x)sech(x) - tanh(x)*-sech(x)tanh(x)\\=&-sech^{3}(x) + tanh^{2}(x)sech(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -sech^{3}(x) + tanh^{2}(x)sech(x)\right)}{dx}\\=&--3sech^{2}(x)sech(x)tanh(x) + 2tanh(x)sech^{2}(x)sech(x) + tanh^{2}(x)*-sech(x)tanh(x)\\=&5tanh(x)sech^{3}(x) - tanh^{3}(x)sech(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 5tanh(x)sech^{3}(x) - tanh^{3}(x)sech(x)\right)}{dx}\\=&5sech^{2}(x)sech^{3}(x) + 5tanh(x)*-3sech^{2}(x)sech(x)tanh(x) - 3tanh^{2}(x)sech^{2}(x)sech(x) - tanh^{3}(x)*-sech(x)tanh(x)\\=&5sech^{5}(x) - 18tanh^{2}(x)sech^{3}(x) + tanh^{4}(x)sech(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/6]Find\ the\ 4th\ derivative\ of\ function\ csch(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( csch(x)\right)}{dx}\\=&-csch(x)coth(x)\\=&-coth(x)csch(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -coth(x)csch(x)\right)}{dx}\\=&--csch^{2}(x)csch(x) - coth(x)*-csch(x)coth(x)\\=&csch^{3}(x) + coth^{2}(x)csch(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( csch^{3}(x) + coth^{2}(x)csch(x)\right)}{dx}\\=&-3csch^{2}(x)csch(x)coth(x) + -2coth(x)csch^{2}(x)csch(x) + coth^{2}(x)*-csch(x)coth(x)\\=&-5coth(x)csch^{3}(x) - coth^{3}(x)csch(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -5coth(x)csch^{3}(x) - coth^{3}(x)csch(x)\right)}{dx}\\=&-5*-csch^{2}(x)csch^{3}(x) - 5coth(x)*-3csch^{2}(x)csch(x)coth(x) - -3coth^{2}(x)csch^{2}(x)csch(x) - coth^{3}(x)*-csch(x)coth(x)\\=&5csch^{5}(x) + 18coth^{2}(x)csch^{3}(x) + coth^{4}(x)csch(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/6]Find\ the\ 4th\ derivative\ of\ function\ tanh(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( tanh(x)\right)}{dx}\\=&sech^{2}(x)\\=&sech^{2}(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( sech^{2}(x)\right)}{dx}\\=&-2sech(x)sech(x)tanh(x)\\=&-2tanh(x)sech^{2}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -2tanh(x)sech^{2}(x)\right)}{dx}\\=&-2sech^{2}(x)sech^{2}(x) - 2tanh(x)*-2sech(x)sech(x)tanh(x)\\=&-2sech^{4}(x) + 4tanh^{2}(x)sech^{2}(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -2sech^{4}(x) + 4tanh^{2}(x)sech^{2}(x)\right)}{dx}\\=&-2*-4sech^{3}(x)sech(x)tanh(x) + 4*2tanh(x)sech^{2}(x)sech^{2}(x) + 4tanh^{2}(x)*-2sech(x)sech(x)tanh(x)\\=&16tanh(x)sech^{4}(x) - 8tanh^{3}(x)sech^{2}(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[4/6]Find\ the\ 4th\ derivative\ of\ function\ th(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( th(x)\right)}{dx}\\=&(1 - th^{2}(x))\\=& - th^{2}(x) + 1\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( - th^{2}(x) + 1\right)}{dx}\\=& - 2th(x)(1 - th^{2}(x)) + 0\\=& - 2th(x) + 2th^{3}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( - 2th(x) + 2th^{3}(x)\right)}{dx}\\=& - 2(1 - th^{2}(x)) + 2*3th^{2}(x)(1 - th^{2}(x))\\=&8th^{2}(x) - 6th^{4}(x) - 2\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 8th^{2}(x) - 6th^{4}(x) - 2\right)}{dx}\\=&8*2th(x)(1 - th^{2}(x)) - 6*4th^{3}(x)(1 - th^{2}(x)) + 0\\=&16th(x) - 40th^{3}(x) + 24th^{5}(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[5/6]Find\ the\ 4th\ derivative\ of\ function\ coth(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( coth(x)\right)}{dx}\\=&-csch^{2}(x)\\=&-csch^{2}(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -csch^{2}(x)\right)}{dx}\\=&--2csch(x)csch(x)coth(x)\\=&2coth(x)csch^{2}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2coth(x)csch^{2}(x)\right)}{dx}\\=&2*-csch^{2}(x)csch^{2}(x) + 2coth(x)*-2csch(x)csch(x)coth(x)\\=&-2csch^{4}(x) - 4coth^{2}(x)csch^{2}(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -2csch^{4}(x) - 4coth^{2}(x)csch^{2}(x)\right)}{dx}\\=&-2*-4csch^{3}(x)csch(x)coth(x) - 4*-2coth(x)csch^{2}(x)csch^{2}(x) - 4coth^{2}(x)*-2csch(x)csch(x)coth(x)\\=&16coth(x)csch^{4}(x) + 8coth^{3}(x)csch^{2}(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[6/6]Find\ the\ 4th\ derivative\ of\ function\ cth(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cth(x)\right)}{dx}\\=&(1 - cth^{2}(x))\\=& - cth^{2}(x) + 1\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( - cth^{2}(x) + 1\right)}{dx}\\=& - 2cth(x)(1 - cth^{2}(x)) + 0\\=& - 2cth(x) + 2cth^{3}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( - 2cth(x) + 2cth^{3}(x)\right)}{dx}\\=& - 2(1 - cth^{2}(x)) + 2*3cth^{2}(x)(1 - cth^{2}(x))\\=&8cth^{2}(x) - 6cth^{4}(x) - 2\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 8cth^{2}(x) - 6cth^{4}(x) - 2\right)}{dx}\\=&8*2cth(x)(1 - cth^{2}(x)) - 6*4cth^{3}(x)(1 - cth^{2}(x)) + 0\\=&16cth(x) - 40cth^{3}(x) + 24cth^{5}(x)\\ \end{split}\end{equation} \]