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 3 questions in this calculation: for each question, the 4 derivative of x is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/3]Find\ the\ 4th\ derivative\ of\ function\ e^{sin(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{sin(x)}\right)}{dx}\\=&e^{sin(x)}cos(x)\\=&e^{sin(x)}cos(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( e^{sin(x)}cos(x)\right)}{dx}\\=&e^{sin(x)}cos(x)cos(x) + e^{sin(x)}*-sin(x)\\=&e^{sin(x)}cos^{2}(x) - e^{sin(x)}sin(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( e^{sin(x)}cos^{2}(x) - e^{sin(x)}sin(x)\right)}{dx}\\=&e^{sin(x)}cos(x)cos^{2}(x) + e^{sin(x)}*-2cos(x)sin(x) - e^{sin(x)}cos(x)sin(x) - e^{sin(x)}cos(x)\\=&e^{sin(x)}cos^{3}(x) - 3e^{sin(x)}sin(x)cos(x) - e^{sin(x)}cos(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( e^{sin(x)}cos^{3}(x) - 3e^{sin(x)}sin(x)cos(x) - e^{sin(x)}cos(x)\right)}{dx}\\=&e^{sin(x)}cos(x)cos^{3}(x) + e^{sin(x)}*-3cos^{2}(x)sin(x) - 3e^{sin(x)}cos(x)sin(x)cos(x) - 3e^{sin(x)}cos(x)cos(x) - 3e^{sin(x)}sin(x)*-sin(x) - e^{sin(x)}cos(x)cos(x) - e^{sin(x)}*-sin(x)\\=&e^{sin(x)}cos^{4}(x) - 6e^{sin(x)}sin(x)cos^{2}(x) - 4e^{sin(x)}cos^{2}(x) + 3e^{sin(x)}sin^{2}(x) + e^{sin(x)}sin(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/3]Find\ the\ 4th\ derivative\ of\ function\ e^{cos(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{cos(x)}\right)}{dx}\\=&e^{cos(x)}*-sin(x)\\=&-e^{cos(x)}sin(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -e^{cos(x)}sin(x)\right)}{dx}\\=&-e^{cos(x)}*-sin(x)sin(x) - e^{cos(x)}cos(x)\\=&e^{cos(x)}sin^{2}(x) - e^{cos(x)}cos(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( e^{cos(x)}sin^{2}(x) - e^{cos(x)}cos(x)\right)}{dx}\\=&e^{cos(x)}*-sin(x)sin^{2}(x) + e^{cos(x)}*2sin(x)cos(x) - e^{cos(x)}*-sin(x)cos(x) - e^{cos(x)}*-sin(x)\\=&3e^{cos(x)}sin(x)cos(x) - e^{cos(x)}sin^{3}(x) + e^{cos(x)}sin(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 3e^{cos(x)}sin(x)cos(x) - e^{cos(x)}sin^{3}(x) + e^{cos(x)}sin(x)\right)}{dx}\\=&3e^{cos(x)}*-sin(x)sin(x)cos(x) + 3e^{cos(x)}cos(x)cos(x) + 3e^{cos(x)}sin(x)*-sin(x) - e^{cos(x)}*-sin(x)sin^{3}(x) - e^{cos(x)}*3sin^{2}(x)cos(x) + e^{cos(x)}*-sin(x)sin(x) + e^{cos(x)}cos(x)\\=&-6e^{cos(x)}sin^{2}(x)cos(x) + 3e^{cos(x)}cos^{2}(x) - 4e^{cos(x)}sin^{2}(x) + e^{cos(x)}sin^{4}(x) + e^{cos(x)}cos(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/3]Find\ the\ 4th\ derivative\ of\ function\ e^{tan(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{tan(x)}\right)}{dx}\\=&e^{tan(x)}sec^{2}(x)(1)\\=&e^{tan(x)}sec^{2}(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( e^{tan(x)}sec^{2}(x)\right)}{dx}\\=&e^{tan(x)}sec^{2}(x)(1)sec^{2}(x) + e^{tan(x)}*2sec^{2}(x)tan(x)\\=&e^{tan(x)}sec^{4}(x) + 2e^{tan(x)}tan(x)sec^{2}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( e^{tan(x)}sec^{4}(x) + 2e^{tan(x)}tan(x)sec^{2}(x)\right)}{dx}\\=&e^{tan(x)}sec^{2}(x)(1)sec^{4}(x) + e^{tan(x)}*4sec^{4}(x)tan(x) + 2e^{tan(x)}sec^{2}(x)(1)tan(x)sec^{2}(x) + 2e^{tan(x)}sec^{2}(x)(1)sec^{2}(x) + 2e^{tan(x)}tan(x)*2sec^{2}(x)tan(x)\\=&e^{tan(x)}sec^{6}(x) + 6e^{tan(x)}tan(x)sec^{4}(x) + 2e^{tan(x)}sec^{4}(x) + 4e^{tan(x)}tan^{2}(x)sec^{2}(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( e^{tan(x)}sec^{6}(x) + 6e^{tan(x)}tan(x)sec^{4}(x) + 2e^{tan(x)}sec^{4}(x) + 4e^{tan(x)}tan^{2}(x)sec^{2}(x)\right)}{dx}\\=&e^{tan(x)}sec^{2}(x)(1)sec^{6}(x) + e^{tan(x)}*6sec^{6}(x)tan(x) + 6e^{tan(x)}sec^{2}(x)(1)tan(x)sec^{4}(x) + 6e^{tan(x)}sec^{2}(x)(1)sec^{4}(x) + 6e^{tan(x)}tan(x)*4sec^{4}(x)tan(x) + 2e^{tan(x)}sec^{2}(x)(1)sec^{4}(x) + 2e^{tan(x)}*4sec^{4}(x)tan(x) + 4e^{tan(x)}sec^{2}(x)(1)tan^{2}(x)sec^{2}(x) + 4e^{tan(x)}*2tan(x)sec^{2}(x)(1)sec^{2}(x) + 4e^{tan(x)}tan^{2}(x)*2sec^{2}(x)tan(x)\\=&e^{tan(x)}sec^{8}(x) + 12e^{tan(x)}tan(x)sec^{6}(x) + 28e^{tan(x)}tan^{2}(x)sec^{4}(x) + 8e^{tan(x)}sec^{6}(x) + 16e^{tan(x)}tan(x)sec^{4}(x) + 8e^{tan(x)}tan^{3}(x)sec^{2}(x)\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/3]Find\ the\ 4th\ derivative\ of\ function\ e^{sin(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{sin(x)}\right)}{dx}\\=&e^{sin(x)}cos(x)\\=&e^{sin(x)}cos(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( e^{sin(x)}cos(x)\right)}{dx}\\=&e^{sin(x)}cos(x)cos(x) + e^{sin(x)}*-sin(x)\\=&e^{sin(x)}cos^{2}(x) - e^{sin(x)}sin(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( e^{sin(x)}cos^{2}(x) - e^{sin(x)}sin(x)\right)}{dx}\\=&e^{sin(x)}cos(x)cos^{2}(x) + e^{sin(x)}*-2cos(x)sin(x) - e^{sin(x)}cos(x)sin(x) - e^{sin(x)}cos(x)\\=&e^{sin(x)}cos^{3}(x) - 3e^{sin(x)}sin(x)cos(x) - e^{sin(x)}cos(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( e^{sin(x)}cos^{3}(x) - 3e^{sin(x)}sin(x)cos(x) - e^{sin(x)}cos(x)\right)}{dx}\\=&e^{sin(x)}cos(x)cos^{3}(x) + e^{sin(x)}*-3cos^{2}(x)sin(x) - 3e^{sin(x)}cos(x)sin(x)cos(x) - 3e^{sin(x)}cos(x)cos(x) - 3e^{sin(x)}sin(x)*-sin(x) - e^{sin(x)}cos(x)cos(x) - e^{sin(x)}*-sin(x)\\=&e^{sin(x)}cos^{4}(x) - 6e^{sin(x)}sin(x)cos^{2}(x) - 4e^{sin(x)}cos^{2}(x) + 3e^{sin(x)}sin^{2}(x) + e^{sin(x)}sin(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[2/3]Find\ the\ 4th\ derivative\ of\ function\ e^{cos(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{cos(x)}\right)}{dx}\\=&e^{cos(x)}*-sin(x)\\=&-e^{cos(x)}sin(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( -e^{cos(x)}sin(x)\right)}{dx}\\=&-e^{cos(x)}*-sin(x)sin(x) - e^{cos(x)}cos(x)\\=&e^{cos(x)}sin^{2}(x) - e^{cos(x)}cos(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( e^{cos(x)}sin^{2}(x) - e^{cos(x)}cos(x)\right)}{dx}\\=&e^{cos(x)}*-sin(x)sin^{2}(x) + e^{cos(x)}*2sin(x)cos(x) - e^{cos(x)}*-sin(x)cos(x) - e^{cos(x)}*-sin(x)\\=&3e^{cos(x)}sin(x)cos(x) - e^{cos(x)}sin^{3}(x) + e^{cos(x)}sin(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 3e^{cos(x)}sin(x)cos(x) - e^{cos(x)}sin^{3}(x) + e^{cos(x)}sin(x)\right)}{dx}\\=&3e^{cos(x)}*-sin(x)sin(x)cos(x) + 3e^{cos(x)}cos(x)cos(x) + 3e^{cos(x)}sin(x)*-sin(x) - e^{cos(x)}*-sin(x)sin^{3}(x) - e^{cos(x)}*3sin^{2}(x)cos(x) + e^{cos(x)}*-sin(x)sin(x) + e^{cos(x)}cos(x)\\=&-6e^{cos(x)}sin^{2}(x)cos(x) + 3e^{cos(x)}cos^{2}(x) - 4e^{cos(x)}sin^{2}(x) + e^{cos(x)}sin^{4}(x) + e^{cos(x)}cos(x)\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}[3/3]Find\ the\ 4th\ derivative\ of\ function\ e^{tan(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{tan(x)}\right)}{dx}\\=&e^{tan(x)}sec^{2}(x)(1)\\=&e^{tan(x)}sec^{2}(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( e^{tan(x)}sec^{2}(x)\right)}{dx}\\=&e^{tan(x)}sec^{2}(x)(1)sec^{2}(x) + e^{tan(x)}*2sec^{2}(x)tan(x)\\=&e^{tan(x)}sec^{4}(x) + 2e^{tan(x)}tan(x)sec^{2}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( e^{tan(x)}sec^{4}(x) + 2e^{tan(x)}tan(x)sec^{2}(x)\right)}{dx}\\=&e^{tan(x)}sec^{2}(x)(1)sec^{4}(x) + e^{tan(x)}*4sec^{4}(x)tan(x) + 2e^{tan(x)}sec^{2}(x)(1)tan(x)sec^{2}(x) + 2e^{tan(x)}sec^{2}(x)(1)sec^{2}(x) + 2e^{tan(x)}tan(x)*2sec^{2}(x)tan(x)\\=&e^{tan(x)}sec^{6}(x) + 6e^{tan(x)}tan(x)sec^{4}(x) + 2e^{tan(x)}sec^{4}(x) + 4e^{tan(x)}tan^{2}(x)sec^{2}(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( e^{tan(x)}sec^{6}(x) + 6e^{tan(x)}tan(x)sec^{4}(x) + 2e^{tan(x)}sec^{4}(x) + 4e^{tan(x)}tan^{2}(x)sec^{2}(x)\right)}{dx}\\=&e^{tan(x)}sec^{2}(x)(1)sec^{6}(x) + e^{tan(x)}*6sec^{6}(x)tan(x) + 6e^{tan(x)}sec^{2}(x)(1)tan(x)sec^{4}(x) + 6e^{tan(x)}sec^{2}(x)(1)sec^{4}(x) + 6e^{tan(x)}tan(x)*4sec^{4}(x)tan(x) + 2e^{tan(x)}sec^{2}(x)(1)sec^{4}(x) + 2e^{tan(x)}*4sec^{4}(x)tan(x) + 4e^{tan(x)}sec^{2}(x)(1)tan^{2}(x)sec^{2}(x) + 4e^{tan(x)}*2tan(x)sec^{2}(x)(1)sec^{2}(x) + 4e^{tan(x)}tan^{2}(x)*2sec^{2}(x)tan(x)\\=&e^{tan(x)}sec^{8}(x) + 12e^{tan(x)}tan(x)sec^{6}(x) + 28e^{tan(x)}tan^{2}(x)sec^{4}(x) + 8e^{tan(x)}sec^{6}(x) + 16e^{tan(x)}tan(x)sec^{4}(x) + 8e^{tan(x)}tan^{3}(x)sec^{2}(x)\\ \end{split}\end{equation} \]