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.
\[ \begin{equation}\begin{split}[1/6]Find\ the\ 4th\ derivative\ of\ function\ sin(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin(x)\right)}{dx}\\=&cos(x)\\=&cos(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( cos(x)\right)}{dx}\\=&-sin(x)\\=&-sin(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -sin(x)\right)}{dx}\\=&-cos(x)\\=&-cos(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -cos(x)\right)}{dx}\\=&--sin(x)\\=&sin(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/6]Find\ the\ 4th\ derivative\ of\ function\ cos(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cos(x)\right)}{dx}\\=&-sin(x)\\=&-sin(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -sin(x)\right)}{dx}\\=&-cos(x)\\=&-cos(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -cos(x)\right)}{dx}\\=&--sin(x)\\=&sin(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( sin(x)\right)}{dx}\\=&cos(x)\\=&cos(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/6]Find\ the\ 4th\ derivative\ of\ function\ tan(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( tan(x)\right)}{dx}\\=&sec^{2}(x)(1)\\=&sec^{2}(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( sec^{2}(x)\right)}{dx}\\=&2sec^{2}(x)tan(x)\\=&2tan(x)sec^{2}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2tan(x)sec^{2}(x)\right)}{dx}\\=&2sec^{2}(x)(1)sec^{2}(x) + 2tan(x)*2sec^{2}(x)tan(x)\\=&2sec^{4}(x) + 4tan^{2}(x)sec^{2}(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 2sec^{4}(x) + 4tan^{2}(x)sec^{2}(x)\right)}{dx}\\=&2*4sec^{4}(x)tan(x) + 4*2tan(x)sec^{2}(x)(1)sec^{2}(x) + 4tan^{2}(x)*2sec^{2}(x)tan(x)\\=&16tan(x)sec^{4}(x) + 8tan^{3}(x)sec^{2}(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[4/6]Find\ the\ 4th\ derivative\ of\ function\ cot(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cot(x)\right)}{dx}\\=&-csc^{2}(x)\\=&-csc^{2}(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -csc^{2}(x)\right)}{dx}\\=&--2csc^{2}(x)cot(x)\\=&2cot(x)csc^{2}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2cot(x)csc^{2}(x)\right)}{dx}\\=&2*-csc^{2}(x)csc^{2}(x) + 2cot(x)*-2csc^{2}(x)cot(x)\\=&-2csc^{4}(x) - 4cot^{2}(x)csc^{2}(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -2csc^{4}(x) - 4cot^{2}(x)csc^{2}(x)\right)}{dx}\\=&-2*-4csc^{4}(x)cot(x) - 4*-2cot(x)csc^{2}(x)csc^{2}(x) - 4cot^{2}(x)*-2csc^{2}(x)cot(x)\\=&16cot(x)csc^{4}(x) + 8cot^{3}(x)csc^{2}(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[5/6]Find\ the\ 4th\ derivative\ of\ function\ sec(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sec(x)\right)}{dx}\\=&sec(x)tan(x)\\=&tan(x)sec(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( tan(x)sec(x)\right)}{dx}\\=&sec^{2}(x)(1)sec(x) + tan(x)sec(x)tan(x)\\=&sec^{3}(x) + tan^{2}(x)sec(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( sec^{3}(x) + tan^{2}(x)sec(x)\right)}{dx}\\=&3sec^{3}(x)tan(x) + 2tan(x)sec^{2}(x)(1)sec(x) + tan^{2}(x)sec(x)tan(x)\\=&5tan(x)sec^{3}(x) + tan^{3}(x)sec(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 5tan(x)sec^{3}(x) + tan^{3}(x)sec(x)\right)}{dx}\\=&5sec^{2}(x)(1)sec^{3}(x) + 5tan(x)*3sec^{3}(x)tan(x) + 3tan^{2}(x)sec^{2}(x)(1)sec(x) + tan^{3}(x)sec(x)tan(x)\\=&5sec^{5}(x) + 18tan^{2}(x)sec^{3}(x) + tan^{4}(x)sec(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[6/6]Find\ the\ 4th\ derivative\ of\ function\ csc(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( csc(x)\right)}{dx}\\=&-csc(x)cot(x)\\=&-cot(x)csc(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -cot(x)csc(x)\right)}{dx}\\=&--csc^{2}(x)csc(x) - cot(x)*-csc(x)cot(x)\\=&csc^{3}(x) + cot^{2}(x)csc(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( csc^{3}(x) + cot^{2}(x)csc(x)\right)}{dx}\\=&-3csc^{3}(x)cot(x) + -2cot(x)csc^{2}(x)csc(x) + cot^{2}(x)*-csc(x)cot(x)\\=& - 5cot(x)csc^{3}(x) - cot^{3}(x)csc(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - 5cot(x)csc^{3}(x) - cot^{3}(x)csc(x)\right)}{dx}\\=& - 5*-csc^{2}(x)csc^{3}(x) - 5cot(x)*-3csc^{3}(x)cot(x) - -3cot^{2}(x)csc^{2}(x)csc(x) - cot^{3}(x)*-csc(x)cot(x)\\=&5csc^{5}(x) + 18cot^{2}(x)csc^{3}(x) + cot^{4}(x)csc(x)\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/6]Find\ the\ 4th\ derivative\ of\ function\ sin(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin(x)\right)}{dx}\\=&cos(x)\\=&cos(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( cos(x)\right)}{dx}\\=&-sin(x)\\=&-sin(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -sin(x)\right)}{dx}\\=&-cos(x)\\=&-cos(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -cos(x)\right)}{dx}\\=&--sin(x)\\=&sin(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/6]Find\ the\ 4th\ derivative\ of\ function\ cos(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cos(x)\right)}{dx}\\=&-sin(x)\\=&-sin(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -sin(x)\right)}{dx}\\=&-cos(x)\\=&-cos(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -cos(x)\right)}{dx}\\=&--sin(x)\\=&sin(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( sin(x)\right)}{dx}\\=&cos(x)\\=&cos(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/6]Find\ the\ 4th\ derivative\ of\ function\ tan(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( tan(x)\right)}{dx}\\=&sec^{2}(x)(1)\\=&sec^{2}(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( sec^{2}(x)\right)}{dx}\\=&2sec^{2}(x)tan(x)\\=&2tan(x)sec^{2}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2tan(x)sec^{2}(x)\right)}{dx}\\=&2sec^{2}(x)(1)sec^{2}(x) + 2tan(x)*2sec^{2}(x)tan(x)\\=&2sec^{4}(x) + 4tan^{2}(x)sec^{2}(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 2sec^{4}(x) + 4tan^{2}(x)sec^{2}(x)\right)}{dx}\\=&2*4sec^{4}(x)tan(x) + 4*2tan(x)sec^{2}(x)(1)sec^{2}(x) + 4tan^{2}(x)*2sec^{2}(x)tan(x)\\=&16tan(x)sec^{4}(x) + 8tan^{3}(x)sec^{2}(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[4/6]Find\ the\ 4th\ derivative\ of\ function\ cot(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cot(x)\right)}{dx}\\=&-csc^{2}(x)\\=&-csc^{2}(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -csc^{2}(x)\right)}{dx}\\=&--2csc^{2}(x)cot(x)\\=&2cot(x)csc^{2}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2cot(x)csc^{2}(x)\right)}{dx}\\=&2*-csc^{2}(x)csc^{2}(x) + 2cot(x)*-2csc^{2}(x)cot(x)\\=&-2csc^{4}(x) - 4cot^{2}(x)csc^{2}(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -2csc^{4}(x) - 4cot^{2}(x)csc^{2}(x)\right)}{dx}\\=&-2*-4csc^{4}(x)cot(x) - 4*-2cot(x)csc^{2}(x)csc^{2}(x) - 4cot^{2}(x)*-2csc^{2}(x)cot(x)\\=&16cot(x)csc^{4}(x) + 8cot^{3}(x)csc^{2}(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[5/6]Find\ the\ 4th\ derivative\ of\ function\ sec(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sec(x)\right)}{dx}\\=&sec(x)tan(x)\\=&tan(x)sec(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( tan(x)sec(x)\right)}{dx}\\=&sec^{2}(x)(1)sec(x) + tan(x)sec(x)tan(x)\\=&sec^{3}(x) + tan^{2}(x)sec(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( sec^{3}(x) + tan^{2}(x)sec(x)\right)}{dx}\\=&3sec^{3}(x)tan(x) + 2tan(x)sec^{2}(x)(1)sec(x) + tan^{2}(x)sec(x)tan(x)\\=&5tan(x)sec^{3}(x) + tan^{3}(x)sec(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 5tan(x)sec^{3}(x) + tan^{3}(x)sec(x)\right)}{dx}\\=&5sec^{2}(x)(1)sec^{3}(x) + 5tan(x)*3sec^{3}(x)tan(x) + 3tan^{2}(x)sec^{2}(x)(1)sec(x) + tan^{3}(x)sec(x)tan(x)\\=&5sec^{5}(x) + 18tan^{2}(x)sec^{3}(x) + tan^{4}(x)sec(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[6/6]Find\ the\ 4th\ derivative\ of\ function\ csc(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( csc(x)\right)}{dx}\\=&-csc(x)cot(x)\\=&-cot(x)csc(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -cot(x)csc(x)\right)}{dx}\\=&--csc^{2}(x)csc(x) - cot(x)*-csc(x)cot(x)\\=&csc^{3}(x) + cot^{2}(x)csc(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( csc^{3}(x) + cot^{2}(x)csc(x)\right)}{dx}\\=&-3csc^{3}(x)cot(x) + -2cot(x)csc^{2}(x)csc(x) + cot^{2}(x)*-csc(x)cot(x)\\=& - 5cot(x)csc^{3}(x) - cot^{3}(x)csc(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - 5cot(x)csc^{3}(x) - cot^{3}(x)csc(x)\right)}{dx}\\=& - 5*-csc^{2}(x)csc^{3}(x) - 5cot(x)*-3csc^{3}(x)cot(x) - -3cot^{2}(x)csc^{2}(x)csc(x) - cot^{3}(x)*-csc(x)cot(x)\\=&5csc^{5}(x) + 18cot^{2}(x)csc^{3}(x) + cot^{4}(x)csc(x)\\ \end{split}\end{equation} \]