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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ e^{e^{e^{sin(x)}}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{e^{e^{sin(x)}}}\right)}{dx}\\=&e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)\\=&e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos(x)\right)}{dx}\\=&e^{sin(x)}cos(x)e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos(x) + e^{sin(x)}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos(x) + e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos(x) + e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}*-sin(x)\\=&e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{2}(x) + e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{2}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}cos^{2}(x) - e^{e^{e^{sin(x)}}}e^{sin(x)}e^{e^{sin(x)}}sin(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{2}(x) + e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{2}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}cos^{2}(x) - e^{e^{e^{sin(x)}}}e^{sin(x)}e^{e^{sin(x)}}sin(x)\right)}{dx}\\=&e^{sin(x)}cos(x)e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{2}(x) + e^{sin(x)}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos^{2}(x) + e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos^{2}(x) + e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}*-2cos(x)sin(x) + 2e^{sin(x)}e^{sin(x)}cos(x)e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{2}(x) + e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos^{2}(x) + e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos^{2}(x) + e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}*-2cos(x)sin(x) + 2e^{e^{sin(x)}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}cos^{2}(x) + e^{{e^{sin(x)}}*{2}}*2e^{sin(x)}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos^{2}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos^{2}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}*-2cos(x)sin(x) - e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{sin(x)}e^{e^{sin(x)}}sin(x) - e^{e^{e^{sin(x)}}}e^{sin(x)}cos(x)e^{e^{sin(x)}}sin(x) - e^{e^{e^{sin(x)}}}e^{sin(x)}e^{e^{sin(x)}}e^{sin(x)}cos(x)sin(x) - e^{e^{e^{sin(x)}}}e^{sin(x)}e^{e^{sin(x)}}cos(x)\\=&e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{3}(x) + 3e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}cos^{3}(x) - 3e^{e^{e^{sin(x)}}}e^{sin(x)}e^{e^{sin(x)}}sin(x)cos(x) + e^{{sin(x)}*{3}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}cos^{3}(x) - 3e^{e^{e^{sin(x)}}}e^{{sin(x)}*{2}}e^{e^{sin(x)}}sin(x)cos(x) + 2e^{{sin(x)}*{3}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}cos^{3}(x) + 2e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{3}}e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}cos^{3}(x) - 2e^{e^{e^{sin(x)}}}e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}sin(x)cos(x) - e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}sin(x)cos(x) - e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{3}(x) + 3e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}cos^{3}(x) - 3e^{e^{e^{sin(x)}}}e^{sin(x)}e^{e^{sin(x)}}sin(x)cos(x) + e^{{sin(x)}*{3}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}cos^{3}(x) - 3e^{e^{e^{sin(x)}}}e^{{sin(x)}*{2}}e^{e^{sin(x)}}sin(x)cos(x) + 2e^{{sin(x)}*{3}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}cos^{3}(x) + 2e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{3}}e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}cos^{3}(x) - 2e^{e^{e^{sin(x)}}}e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}sin(x)cos(x) - e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}sin(x)cos(x) - e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos(x)\right)}{dx}\\=&e^{sin(x)}cos(x)e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{sin(x)}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos^{3}(x) + e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}*-3cos^{2}(x)sin(x) + 3*2e^{sin(x)}e^{sin(x)}cos(x)e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{3}(x) + 3e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos^{3}(x) + 3e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos^{3}(x) + 3e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}*-3cos^{2}(x)sin(x) + 2e^{e^{sin(x)}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{2}}*2e^{sin(x)}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos^{3}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}*-3cos^{2}(x)sin(x) - 3e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{sin(x)}e^{e^{sin(x)}}sin(x)cos(x) - 3e^{e^{e^{sin(x)}}}e^{sin(x)}cos(x)e^{e^{sin(x)}}sin(x)cos(x) - 3e^{e^{e^{sin(x)}}}e^{sin(x)}e^{e^{sin(x)}}e^{sin(x)}cos(x)sin(x)cos(x) - 3e^{e^{e^{sin(x)}}}e^{sin(x)}e^{e^{sin(x)}}cos(x)cos(x) - 3e^{e^{e^{sin(x)}}}e^{sin(x)}e^{e^{sin(x)}}sin(x)*-sin(x) + 3e^{{sin(x)}*{2}}e^{sin(x)}cos(x)e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{sin(x)}*{3}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{sin(x)}*{3}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos^{3}(x) + e^{{sin(x)}*{3}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}*-3cos^{2}(x)sin(x) + 2e^{e^{sin(x)}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{2}}*3e^{{sin(x)}*{2}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos^{3}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}*-3cos^{2}(x)sin(x) - 3e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{{sin(x)}*{2}}e^{e^{sin(x)}}sin(x)cos(x) - 3e^{e^{e^{sin(x)}}}*2e^{sin(x)}e^{sin(x)}cos(x)e^{e^{sin(x)}}sin(x)cos(x) - 3e^{e^{e^{sin(x)}}}e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{sin(x)}cos(x)sin(x)cos(x) - 3e^{e^{e^{sin(x)}}}e^{{sin(x)}*{2}}e^{e^{sin(x)}}cos(x)cos(x) - 3e^{e^{e^{sin(x)}}}e^{{sin(x)}*{2}}e^{e^{sin(x)}}sin(x)*-sin(x) + 2*3e^{{sin(x)}*{2}}e^{sin(x)}cos(x)e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}cos^{3}(x) + 2e^{{sin(x)}*{3}}*2e^{e^{sin(x)}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos^{3}(x) + 2e^{{sin(x)}*{3}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos^{3}(x) + 2e^{{sin(x)}*{3}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}*-3cos^{2}(x)sin(x) + 2*2e^{sin(x)}e^{sin(x)}cos(x)e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}cos^{3}(x) + 2e^{{sin(x)}*{2}}*2e^{e^{sin(x)}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos^{3}(x) + 2e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos^{3}(x) + 2e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}*-3cos^{2}(x)sin(x) + 3e^{{e^{sin(x)}}*{2}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{3}}*3e^{{sin(x)}*{2}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{3}}e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos^{3}(x) + e^{{e^{sin(x)}}*{3}}e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}*-3cos^{2}(x)sin(x) - 2e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}sin(x)cos(x) - 2e^{e^{e^{sin(x)}}}*2e^{e^{sin(x)}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{{sin(x)}*{2}}sin(x)cos(x) - 2e^{e^{e^{sin(x)}}}e^{{e^{sin(x)}}*{2}}*2e^{sin(x)}e^{sin(x)}cos(x)sin(x)cos(x) - 2e^{e^{e^{sin(x)}}}e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}cos(x)cos(x) - 2e^{e^{e^{sin(x)}}}e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}sin(x)*-sin(x) - 2e^{sin(x)}e^{sin(x)}cos(x)e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}sin(x)cos(x) - e^{{sin(x)}*{2}}*2e^{e^{sin(x)}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}sin(x)cos(x) - e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)sin(x)cos(x) - e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}cos(x)cos(x) - e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}sin(x)*-sin(x) - e^{sin(x)}cos(x)e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos(x) - e^{sin(x)}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos(x) - e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos(x) - e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}*-sin(x)\\=&e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{4}(x) + 7e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{4}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}cos^{4}(x) - 6e^{e^{e^{sin(x)}}}e^{sin(x)}e^{e^{sin(x)}}sin(x)cos^{2}(x) + 6e^{{sin(x)}*{3}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{4}(x) + 3e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}cos^{4}(x) - 12e^{e^{e^{sin(x)}}}e^{{sin(x)}*{2}}e^{e^{sin(x)}}sin(x)cos^{2}(x) + 15e^{{sin(x)}*{3}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}cos^{4}(x) + 6e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}cos^{4}(x) + 3e^{{e^{sin(x)}}*{3}}e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}cos^{4}(x) - 3e^{e^{e^{sin(x)}}}e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}sin(x)cos^{2}(x) - 9e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}sin(x)cos^{2}(x) - 4e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{2}(x) - 6e^{e^{e^{sin(x)}}}e^{{sin(x)}*{3}}e^{e^{sin(x)}}sin(x)cos^{2}(x) + e^{{sin(x)}*{4}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{4}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{4}}e^{e^{e^{sin(x)}}}cos^{4}(x) - 3e^{e^{e^{sin(x)}}}e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{3}}sin(x)cos^{2}(x) + 6e^{{sin(x)}*{4}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}cos^{4}(x) + 3e^{{e^{sin(x)}}*{3}}e^{{sin(x)}*{4}}e^{e^{e^{sin(x)}}}cos^{4}(x) - 5e^{{sin(x)}*{3}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}sin(x)cos^{2}(x) - 6e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}sin(x)cos^{2}(x) - 10e^{e^{e^{sin(x)}}}e^{{sin(x)}*{3}}e^{{e^{sin(x)}}*{2}}sin(x)cos^{2}(x) - 4e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{2}(x) - 6e^{e^{e^{sin(x)}}}e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}sin(x)cos^{2}(x) - 3e^{e^{e^{sin(x)}}}e^{{e^{sin(x)}}*{3}}e^{{sin(x)}*{3}}sin(x)cos^{2}(x) - 2e^{{sin(x)}*{3}}e^{{e^{sin(x)}}*{3}}e^{e^{e^{sin(x)}}}sin(x)cos^{2}(x) + 3e^{{sin(x)}*{4}}e^{{e^{sin(x)}}*{3}}e^{e^{e^{sin(x)}}}cos^{4}(x) + 3e^{{sin(x)}*{3}}e^{{e^{sin(x)}}*{3}}e^{e^{e^{sin(x)}}}cos^{4}(x) + e^{{e^{sin(x)}}*{4}}e^{{sin(x)}*{4}}e^{e^{e^{sin(x)}}}cos^{4}(x) - e^{{e^{sin(x)}}*{3}}e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}sin(x)cos^{2}(x) + 3e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}e^{{sin(x)}*{2}}sin^{2}(x) - 3e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}cos^{2}(x) + 2e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}e^{{e^{sin(x)}}*{2}}sin^{2}(x) + 3e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}e^{sin(x)}sin^{2}(x) - e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}e^{{sin(x)}*{2}}cos^{2}(x) + e^{e^{e^{sin(x)}}}e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}sin^{2}(x) + e^{e^{e^{sin(x)}}}e^{sin(x)}e^{e^{sin(x)}}sin(x)\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ e^{e^{e^{sin(x)}}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{e^{e^{sin(x)}}}\right)}{dx}\\=&e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)\\=&e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos(x)\right)}{dx}\\=&e^{sin(x)}cos(x)e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos(x) + e^{sin(x)}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos(x) + e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos(x) + e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}*-sin(x)\\=&e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{2}(x) + e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{2}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}cos^{2}(x) - e^{e^{e^{sin(x)}}}e^{sin(x)}e^{e^{sin(x)}}sin(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{2}(x) + e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{2}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}cos^{2}(x) - e^{e^{e^{sin(x)}}}e^{sin(x)}e^{e^{sin(x)}}sin(x)\right)}{dx}\\=&e^{sin(x)}cos(x)e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{2}(x) + e^{sin(x)}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos^{2}(x) + e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos^{2}(x) + e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}*-2cos(x)sin(x) + 2e^{sin(x)}e^{sin(x)}cos(x)e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{2}(x) + e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos^{2}(x) + e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos^{2}(x) + e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}*-2cos(x)sin(x) + 2e^{e^{sin(x)}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}cos^{2}(x) + e^{{e^{sin(x)}}*{2}}*2e^{sin(x)}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos^{2}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos^{2}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}*-2cos(x)sin(x) - e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{sin(x)}e^{e^{sin(x)}}sin(x) - e^{e^{e^{sin(x)}}}e^{sin(x)}cos(x)e^{e^{sin(x)}}sin(x) - e^{e^{e^{sin(x)}}}e^{sin(x)}e^{e^{sin(x)}}e^{sin(x)}cos(x)sin(x) - e^{e^{e^{sin(x)}}}e^{sin(x)}e^{e^{sin(x)}}cos(x)\\=&e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{3}(x) + 3e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}cos^{3}(x) - 3e^{e^{e^{sin(x)}}}e^{sin(x)}e^{e^{sin(x)}}sin(x)cos(x) + e^{{sin(x)}*{3}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}cos^{3}(x) - 3e^{e^{e^{sin(x)}}}e^{{sin(x)}*{2}}e^{e^{sin(x)}}sin(x)cos(x) + 2e^{{sin(x)}*{3}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}cos^{3}(x) + 2e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{3}}e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}cos^{3}(x) - 2e^{e^{e^{sin(x)}}}e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}sin(x)cos(x) - e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}sin(x)cos(x) - e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{3}(x) + 3e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}cos^{3}(x) - 3e^{e^{e^{sin(x)}}}e^{sin(x)}e^{e^{sin(x)}}sin(x)cos(x) + e^{{sin(x)}*{3}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}cos^{3}(x) - 3e^{e^{e^{sin(x)}}}e^{{sin(x)}*{2}}e^{e^{sin(x)}}sin(x)cos(x) + 2e^{{sin(x)}*{3}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}cos^{3}(x) + 2e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{3}}e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}cos^{3}(x) - 2e^{e^{e^{sin(x)}}}e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}sin(x)cos(x) - e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}sin(x)cos(x) - e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos(x)\right)}{dx}\\=&e^{sin(x)}cos(x)e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{sin(x)}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos^{3}(x) + e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}*-3cos^{2}(x)sin(x) + 3*2e^{sin(x)}e^{sin(x)}cos(x)e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{3}(x) + 3e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos^{3}(x) + 3e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos^{3}(x) + 3e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}*-3cos^{2}(x)sin(x) + 2e^{e^{sin(x)}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{2}}*2e^{sin(x)}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos^{3}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}*-3cos^{2}(x)sin(x) - 3e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{sin(x)}e^{e^{sin(x)}}sin(x)cos(x) - 3e^{e^{e^{sin(x)}}}e^{sin(x)}cos(x)e^{e^{sin(x)}}sin(x)cos(x) - 3e^{e^{e^{sin(x)}}}e^{sin(x)}e^{e^{sin(x)}}e^{sin(x)}cos(x)sin(x)cos(x) - 3e^{e^{e^{sin(x)}}}e^{sin(x)}e^{e^{sin(x)}}cos(x)cos(x) - 3e^{e^{e^{sin(x)}}}e^{sin(x)}e^{e^{sin(x)}}sin(x)*-sin(x) + 3e^{{sin(x)}*{2}}e^{sin(x)}cos(x)e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{sin(x)}*{3}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{sin(x)}*{3}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos^{3}(x) + e^{{sin(x)}*{3}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}*-3cos^{2}(x)sin(x) + 2e^{e^{sin(x)}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{2}}*3e^{{sin(x)}*{2}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos^{3}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}*-3cos^{2}(x)sin(x) - 3e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{{sin(x)}*{2}}e^{e^{sin(x)}}sin(x)cos(x) - 3e^{e^{e^{sin(x)}}}*2e^{sin(x)}e^{sin(x)}cos(x)e^{e^{sin(x)}}sin(x)cos(x) - 3e^{e^{e^{sin(x)}}}e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{sin(x)}cos(x)sin(x)cos(x) - 3e^{e^{e^{sin(x)}}}e^{{sin(x)}*{2}}e^{e^{sin(x)}}cos(x)cos(x) - 3e^{e^{e^{sin(x)}}}e^{{sin(x)}*{2}}e^{e^{sin(x)}}sin(x)*-sin(x) + 2*3e^{{sin(x)}*{2}}e^{sin(x)}cos(x)e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}cos^{3}(x) + 2e^{{sin(x)}*{3}}*2e^{e^{sin(x)}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos^{3}(x) + 2e^{{sin(x)}*{3}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos^{3}(x) + 2e^{{sin(x)}*{3}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}*-3cos^{2}(x)sin(x) + 2*2e^{sin(x)}e^{sin(x)}cos(x)e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}cos^{3}(x) + 2e^{{sin(x)}*{2}}*2e^{e^{sin(x)}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos^{3}(x) + 2e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos^{3}(x) + 2e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}*-3cos^{2}(x)sin(x) + 3e^{{e^{sin(x)}}*{2}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{3}}*3e^{{sin(x)}*{2}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos^{3}(x) + e^{{e^{sin(x)}}*{3}}e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos^{3}(x) + e^{{e^{sin(x)}}*{3}}e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}*-3cos^{2}(x)sin(x) - 2e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}sin(x)cos(x) - 2e^{e^{e^{sin(x)}}}*2e^{e^{sin(x)}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{{sin(x)}*{2}}sin(x)cos(x) - 2e^{e^{e^{sin(x)}}}e^{{e^{sin(x)}}*{2}}*2e^{sin(x)}e^{sin(x)}cos(x)sin(x)cos(x) - 2e^{e^{e^{sin(x)}}}e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}cos(x)cos(x) - 2e^{e^{e^{sin(x)}}}e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}sin(x)*-sin(x) - 2e^{sin(x)}e^{sin(x)}cos(x)e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}sin(x)cos(x) - e^{{sin(x)}*{2}}*2e^{e^{sin(x)}}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}sin(x)cos(x) - e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)sin(x)cos(x) - e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}cos(x)cos(x) - e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}sin(x)*-sin(x) - e^{sin(x)}cos(x)e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos(x) - e^{sin(x)}e^{e^{sin(x)}}e^{sin(x)}cos(x)e^{e^{e^{sin(x)}}}cos(x) - e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}e^{sin(x)}cos(x)cos(x) - e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}*-sin(x)\\=&e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{4}(x) + 7e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{4}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}cos^{4}(x) - 6e^{e^{e^{sin(x)}}}e^{sin(x)}e^{e^{sin(x)}}sin(x)cos^{2}(x) + 6e^{{sin(x)}*{3}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{4}(x) + 3e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}cos^{4}(x) - 12e^{e^{e^{sin(x)}}}e^{{sin(x)}*{2}}e^{e^{sin(x)}}sin(x)cos^{2}(x) + 15e^{{sin(x)}*{3}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}cos^{4}(x) + 6e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}cos^{4}(x) + 3e^{{e^{sin(x)}}*{3}}e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}cos^{4}(x) - 3e^{e^{e^{sin(x)}}}e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}sin(x)cos^{2}(x) - 9e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}sin(x)cos^{2}(x) - 4e^{sin(x)}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{2}(x) - 6e^{e^{e^{sin(x)}}}e^{{sin(x)}*{3}}e^{e^{sin(x)}}sin(x)cos^{2}(x) + e^{{sin(x)}*{4}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{4}(x) + e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{4}}e^{e^{e^{sin(x)}}}cos^{4}(x) - 3e^{e^{e^{sin(x)}}}e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{3}}sin(x)cos^{2}(x) + 6e^{{sin(x)}*{4}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}cos^{4}(x) + 3e^{{e^{sin(x)}}*{3}}e^{{sin(x)}*{4}}e^{e^{e^{sin(x)}}}cos^{4}(x) - 5e^{{sin(x)}*{3}}e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}sin(x)cos^{2}(x) - 6e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}e^{e^{sin(x)}}sin(x)cos^{2}(x) - 10e^{e^{e^{sin(x)}}}e^{{sin(x)}*{3}}e^{{e^{sin(x)}}*{2}}sin(x)cos^{2}(x) - 4e^{{sin(x)}*{2}}e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}cos^{2}(x) - 6e^{e^{e^{sin(x)}}}e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}sin(x)cos^{2}(x) - 3e^{e^{e^{sin(x)}}}e^{{e^{sin(x)}}*{3}}e^{{sin(x)}*{3}}sin(x)cos^{2}(x) - 2e^{{sin(x)}*{3}}e^{{e^{sin(x)}}*{3}}e^{e^{e^{sin(x)}}}sin(x)cos^{2}(x) + 3e^{{sin(x)}*{4}}e^{{e^{sin(x)}}*{3}}e^{e^{e^{sin(x)}}}cos^{4}(x) + 3e^{{sin(x)}*{3}}e^{{e^{sin(x)}}*{3}}e^{e^{e^{sin(x)}}}cos^{4}(x) + e^{{e^{sin(x)}}*{4}}e^{{sin(x)}*{4}}e^{e^{e^{sin(x)}}}cos^{4}(x) - e^{{e^{sin(x)}}*{3}}e^{{sin(x)}*{3}}e^{e^{e^{sin(x)}}}sin(x)cos^{2}(x) + 3e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}e^{{sin(x)}*{2}}sin^{2}(x) - 3e^{{e^{sin(x)}}*{2}}e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}cos^{2}(x) + 2e^{{sin(x)}*{2}}e^{e^{e^{sin(x)}}}e^{{e^{sin(x)}}*{2}}sin^{2}(x) + 3e^{e^{sin(x)}}e^{e^{e^{sin(x)}}}e^{sin(x)}sin^{2}(x) - e^{{e^{sin(x)}}*{2}}e^{e^{e^{sin(x)}}}e^{{sin(x)}*{2}}cos^{2}(x) + e^{e^{e^{sin(x)}}}e^{{sin(x)}*{2}}e^{{e^{sin(x)}}*{2}}sin^{2}(x) + e^{e^{e^{sin(x)}}}e^{sin(x)}e^{e^{sin(x)}}sin(x)\\ \end{split}\end{equation} \]