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 b is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ ({b}^{4} - 6{b}^{3} + 11{b}^{2} - 6b){a}^{b}{\frac{1}{a}}^{4} + sin(a)(cos(b))\ with\ respect\ to\ b：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{b^{4}{a}^{b}}{a^{4}} - \frac{6b^{3}{a}^{b}}{a^{4}} + \frac{11b^{2}{a}^{b}}{a^{4}} - \frac{6b{a}^{b}}{a^{4}} + sin(a)cos(b)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{b^{4}{a}^{b}}{a^{4}} - \frac{6b^{3}{a}^{b}}{a^{4}} + \frac{11b^{2}{a}^{b}}{a^{4}} - \frac{6b{a}^{b}}{a^{4}} + sin(a)cos(b)\right)}{db}\\=&\frac{4b^{3}{a}^{b}}{a^{4}} + \frac{b^{4}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} - \frac{6*3b^{2}{a}^{b}}{a^{4}} - \frac{6b^{3}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} + \frac{11*2b{a}^{b}}{a^{4}} + \frac{11b^{2}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} - \frac{6{a}^{b}}{a^{4}} - \frac{6b({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} + cos(a)*0cos(b) + sin(a)*-sin(b)\\=&\frac{b^{4}{a}^{b}ln(a)}{a^{4}} - \frac{6b^{3}{a}^{b}ln(a)}{a^{4}} + \frac{11b^{2}{a}^{b}ln(a)}{a^{4}} - \frac{6b{a}^{b}ln(a)}{a^{4}} + \frac{22b{a}^{b}}{a^{4}} - \frac{18b^{2}{a}^{b}}{a^{4}} - \frac{6{a}^{b}}{a^{4}} + \frac{4b^{3}{a}^{b}}{a^{4}} - sin(b)sin(a)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{b^{4}{a}^{b}ln(a)}{a^{4}} - \frac{6b^{3}{a}^{b}ln(a)}{a^{4}} + \frac{11b^{2}{a}^{b}ln(a)}{a^{4}} - \frac{6b{a}^{b}ln(a)}{a^{4}} + \frac{22b{a}^{b}}{a^{4}} - \frac{18b^{2}{a}^{b}}{a^{4}} - \frac{6{a}^{b}}{a^{4}} + \frac{4b^{3}{a}^{b}}{a^{4}} - sin(b)sin(a)\right)}{db}\\=&\frac{4b^{3}{a}^{b}ln(a)}{a^{4}} + \frac{b^{4}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln(a)}{a^{4}} + \frac{b^{4}{a}^{b}*0}{a^{4}(a)} - \frac{6*3b^{2}{a}^{b}ln(a)}{a^{4}} - \frac{6b^{3}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln(a)}{a^{4}} - \frac{6b^{3}{a}^{b}*0}{a^{4}(a)} + \frac{11*2b{a}^{b}ln(a)}{a^{4}} + \frac{11b^{2}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln(a)}{a^{4}} + \frac{11b^{2}{a}^{b}*0}{a^{4}(a)} - \frac{6{a}^{b}ln(a)}{a^{4}} - \frac{6b({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln(a)}{a^{4}} - \frac{6b{a}^{b}*0}{a^{4}(a)} + \frac{22{a}^{b}}{a^{4}} + \frac{22b({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} - \frac{18*2b{a}^{b}}{a^{4}} - \frac{18b^{2}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} - \frac{6({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} + \frac{4*3b^{2}{a}^{b}}{a^{4}} + \frac{4b^{3}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} - cos(b)sin(a) - sin(b)cos(a)*0\\=&\frac{8b^{3}{a}^{b}ln(a)}{a^{4}} + \frac{b^{4}{a}^{b}ln^{2}(a)}{a^{4}} - \frac{36b^{2}{a}^{b}ln(a)}{a^{4}} - \frac{6b^{3}{a}^{b}ln^{2}(a)}{a^{4}} + \frac{44b{a}^{b}ln(a)}{a^{4}} + \frac{11b^{2}{a}^{b}ln^{2}(a)}{a^{4}} - \frac{12{a}^{b}ln(a)}{a^{4}} - \frac{6b{a}^{b}ln^{2}(a)}{a^{4}} + \frac{22{a}^{b}}{a^{4}} - \frac{36b{a}^{b}}{a^{4}} + \frac{12b^{2}{a}^{b}}{a^{4}} - sin(a)cos(b)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{8b^{3}{a}^{b}ln(a)}{a^{4}} + \frac{b^{4}{a}^{b}ln^{2}(a)}{a^{4}} - \frac{36b^{2}{a}^{b}ln(a)}{a^{4}} - \frac{6b^{3}{a}^{b}ln^{2}(a)}{a^{4}} + \frac{44b{a}^{b}ln(a)}{a^{4}} + \frac{11b^{2}{a}^{b}ln^{2}(a)}{a^{4}} - \frac{12{a}^{b}ln(a)}{a^{4}} - \frac{6b{a}^{b}ln^{2}(a)}{a^{4}} + \frac{22{a}^{b}}{a^{4}} - \frac{36b{a}^{b}}{a^{4}} + \frac{12b^{2}{a}^{b}}{a^{4}} - sin(a)cos(b)\right)}{db}\\=&\frac{8*3b^{2}{a}^{b}ln(a)}{a^{4}} + \frac{8b^{3}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln(a)}{a^{4}} + \frac{8b^{3}{a}^{b}*0}{a^{4}(a)} + \frac{4b^{3}{a}^{b}ln^{2}(a)}{a^{4}} + \frac{b^{4}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{2}(a)}{a^{4}} + \frac{b^{4}{a}^{b}*2ln(a)*0}{a^{4}(a)} - \frac{36*2b{a}^{b}ln(a)}{a^{4}} - \frac{36b^{2}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln(a)}{a^{4}} - \frac{36b^{2}{a}^{b}*0}{a^{4}(a)} - \frac{6*3b^{2}{a}^{b}ln^{2}(a)}{a^{4}} - \frac{6b^{3}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{2}(a)}{a^{4}} - \frac{6b^{3}{a}^{b}*2ln(a)*0}{a^{4}(a)} + \frac{44{a}^{b}ln(a)}{a^{4}} + \frac{44b({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln(a)}{a^{4}} + \frac{44b{a}^{b}*0}{a^{4}(a)} + \frac{11*2b{a}^{b}ln^{2}(a)}{a^{4}} + \frac{11b^{2}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{2}(a)}{a^{4}} + \frac{11b^{2}{a}^{b}*2ln(a)*0}{a^{4}(a)} - \frac{12({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln(a)}{a^{4}} - \frac{12{a}^{b}*0}{a^{4}(a)} - \frac{6{a}^{b}ln^{2}(a)}{a^{4}} - \frac{6b({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{2}(a)}{a^{4}} - \frac{6b{a}^{b}*2ln(a)*0}{a^{4}(a)} + \frac{22({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} - \frac{36{a}^{b}}{a^{4}} - \frac{36b({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} + \frac{12*2b{a}^{b}}{a^{4}} + \frac{12b^{2}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} - cos(a)*0cos(b) - sin(a)*-sin(b)\\=&\frac{36b^{2}{a}^{b}ln(a)}{a^{4}} + \frac{12b^{3}{a}^{b}ln^{2}(a)}{a^{4}} + \frac{b^{4}{a}^{b}ln^{3}(a)}{a^{4}} - \frac{108b{a}^{b}ln(a)}{a^{4}} - \frac{54b^{2}{a}^{b}ln^{2}(a)}{a^{4}} - \frac{6b^{3}{a}^{b}ln^{3}(a)}{a^{4}} + \frac{66{a}^{b}ln(a)}{a^{4}} + \frac{66b{a}^{b}ln^{2}(a)}{a^{4}} + \frac{11b^{2}{a}^{b}ln^{3}(a)}{a^{4}} - \frac{18{a}^{b}ln^{2}(a)}{a^{4}} - \frac{6b{a}^{b}ln^{3}(a)}{a^{4}} - \frac{36{a}^{b}}{a^{4}} + \frac{24b{a}^{b}}{a^{4}} + sin(b)sin(a)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{36b^{2}{a}^{b}ln(a)}{a^{4}} + \frac{12b^{3}{a}^{b}ln^{2}(a)}{a^{4}} + \frac{b^{4}{a}^{b}ln^{3}(a)}{a^{4}} - \frac{108b{a}^{b}ln(a)}{a^{4}} - \frac{54b^{2}{a}^{b}ln^{2}(a)}{a^{4}} - \frac{6b^{3}{a}^{b}ln^{3}(a)}{a^{4}} + \frac{66{a}^{b}ln(a)}{a^{4}} + \frac{66b{a}^{b}ln^{2}(a)}{a^{4}} + \frac{11b^{2}{a}^{b}ln^{3}(a)}{a^{4}} - \frac{18{a}^{b}ln^{2}(a)}{a^{4}} - \frac{6b{a}^{b}ln^{3}(a)}{a^{4}} - \frac{36{a}^{b}}{a^{4}} + \frac{24b{a}^{b}}{a^{4}} + sin(b)sin(a)\right)}{db}\\=&\frac{36*2b{a}^{b}ln(a)}{a^{4}} + \frac{36b^{2}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln(a)}{a^{4}} + \frac{36b^{2}{a}^{b}*0}{a^{4}(a)} + \frac{12*3b^{2}{a}^{b}ln^{2}(a)}{a^{4}} + \frac{12b^{3}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{2}(a)}{a^{4}} + \frac{12b^{3}{a}^{b}*2ln(a)*0}{a^{4}(a)} + \frac{4b^{3}{a}^{b}ln^{3}(a)}{a^{4}} + \frac{b^{4}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{3}(a)}{a^{4}} + \frac{b^{4}{a}^{b}*3ln^{2}(a)*0}{a^{4}(a)} - \frac{108{a}^{b}ln(a)}{a^{4}} - \frac{108b({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln(a)}{a^{4}} - \frac{108b{a}^{b}*0}{a^{4}(a)} - \frac{54*2b{a}^{b}ln^{2}(a)}{a^{4}} - \frac{54b^{2}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{2}(a)}{a^{4}} - \frac{54b^{2}{a}^{b}*2ln(a)*0}{a^{4}(a)} - \frac{6*3b^{2}{a}^{b}ln^{3}(a)}{a^{4}} - \frac{6b^{3}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{3}(a)}{a^{4}} - \frac{6b^{3}{a}^{b}*3ln^{2}(a)*0}{a^{4}(a)} + \frac{66({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln(a)}{a^{4}} + \frac{66{a}^{b}*0}{a^{4}(a)} + \frac{66{a}^{b}ln^{2}(a)}{a^{4}} + \frac{66b({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{2}(a)}{a^{4}} + \frac{66b{a}^{b}*2ln(a)*0}{a^{4}(a)} + \frac{11*2b{a}^{b}ln^{3}(a)}{a^{4}} + \frac{11b^{2}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{3}(a)}{a^{4}} + \frac{11b^{2}{a}^{b}*3ln^{2}(a)*0}{a^{4}(a)} - \frac{18({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{2}(a)}{a^{4}} - \frac{18{a}^{b}*2ln(a)*0}{a^{4}(a)} - \frac{6{a}^{b}ln^{3}(a)}{a^{4}} - \frac{6b({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{3}(a)}{a^{4}} - \frac{6b{a}^{b}*3ln^{2}(a)*0}{a^{4}(a)} - \frac{36({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} + \frac{24{a}^{b}}{a^{4}} + \frac{24b({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} + cos(b)sin(a) + sin(b)cos(a)*0\\=&\frac{96b{a}^{b}ln(a)}{a^{4}} + \frac{72b^{2}{a}^{b}ln^{2}(a)}{a^{4}} + \frac{16b^{3}{a}^{b}ln^{3}(a)}{a^{4}} + \frac{b^{4}{a}^{b}ln^{4}(a)}{a^{4}} - \frac{144{a}^{b}ln(a)}{a^{4}} - \frac{216b{a}^{b}ln^{2}(a)}{a^{4}} - \frac{72b^{2}{a}^{b}ln^{3}(a)}{a^{4}} - \frac{6b^{3}{a}^{b}ln^{4}(a)}{a^{4}} + \frac{132{a}^{b}ln^{2}(a)}{a^{4}} + \frac{88b{a}^{b}ln^{3}(a)}{a^{4}} + \frac{11b^{2}{a}^{b}ln^{4}(a)}{a^{4}} - \frac{24{a}^{b}ln^{3}(a)}{a^{4}} - \frac{6b{a}^{b}ln^{4}(a)}{a^{4}} + \frac{24{a}^{b}}{a^{4}} + sin(a)cos(b)\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ ({b}^{4} - 6{b}^{3} + 11{b}^{2} - 6b){a}^{b}{\frac{1}{a}}^{4} + sin(a)(cos(b))\ with\ respect\ to\ b：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{b^{4}{a}^{b}}{a^{4}} - \frac{6b^{3}{a}^{b}}{a^{4}} + \frac{11b^{2}{a}^{b}}{a^{4}} - \frac{6b{a}^{b}}{a^{4}} + sin(a)cos(b)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{b^{4}{a}^{b}}{a^{4}} - \frac{6b^{3}{a}^{b}}{a^{4}} + \frac{11b^{2}{a}^{b}}{a^{4}} - \frac{6b{a}^{b}}{a^{4}} + sin(a)cos(b)\right)}{db}\\=&\frac{4b^{3}{a}^{b}}{a^{4}} + \frac{b^{4}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} - \frac{6*3b^{2}{a}^{b}}{a^{4}} - \frac{6b^{3}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} + \frac{11*2b{a}^{b}}{a^{4}} + \frac{11b^{2}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} - \frac{6{a}^{b}}{a^{4}} - \frac{6b({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} + cos(a)*0cos(b) + sin(a)*-sin(b)\\=&\frac{b^{4}{a}^{b}ln(a)}{a^{4}} - \frac{6b^{3}{a}^{b}ln(a)}{a^{4}} + \frac{11b^{2}{a}^{b}ln(a)}{a^{4}} - \frac{6b{a}^{b}ln(a)}{a^{4}} + \frac{22b{a}^{b}}{a^{4}} - \frac{18b^{2}{a}^{b}}{a^{4}} - \frac{6{a}^{b}}{a^{4}} + \frac{4b^{3}{a}^{b}}{a^{4}} - sin(b)sin(a)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{b^{4}{a}^{b}ln(a)}{a^{4}} - \frac{6b^{3}{a}^{b}ln(a)}{a^{4}} + \frac{11b^{2}{a}^{b}ln(a)}{a^{4}} - \frac{6b{a}^{b}ln(a)}{a^{4}} + \frac{22b{a}^{b}}{a^{4}} - \frac{18b^{2}{a}^{b}}{a^{4}} - \frac{6{a}^{b}}{a^{4}} + \frac{4b^{3}{a}^{b}}{a^{4}} - sin(b)sin(a)\right)}{db}\\=&\frac{4b^{3}{a}^{b}ln(a)}{a^{4}} + \frac{b^{4}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln(a)}{a^{4}} + \frac{b^{4}{a}^{b}*0}{a^{4}(a)} - \frac{6*3b^{2}{a}^{b}ln(a)}{a^{4}} - \frac{6b^{3}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln(a)}{a^{4}} - \frac{6b^{3}{a}^{b}*0}{a^{4}(a)} + \frac{11*2b{a}^{b}ln(a)}{a^{4}} + \frac{11b^{2}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln(a)}{a^{4}} + \frac{11b^{2}{a}^{b}*0}{a^{4}(a)} - \frac{6{a}^{b}ln(a)}{a^{4}} - \frac{6b({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln(a)}{a^{4}} - \frac{6b{a}^{b}*0}{a^{4}(a)} + \frac{22{a}^{b}}{a^{4}} + \frac{22b({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} - \frac{18*2b{a}^{b}}{a^{4}} - \frac{18b^{2}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} - \frac{6({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} + \frac{4*3b^{2}{a}^{b}}{a^{4}} + \frac{4b^{3}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} - cos(b)sin(a) - sin(b)cos(a)*0\\=&\frac{8b^{3}{a}^{b}ln(a)}{a^{4}} + \frac{b^{4}{a}^{b}ln^{2}(a)}{a^{4}} - \frac{36b^{2}{a}^{b}ln(a)}{a^{4}} - \frac{6b^{3}{a}^{b}ln^{2}(a)}{a^{4}} + \frac{44b{a}^{b}ln(a)}{a^{4}} + \frac{11b^{2}{a}^{b}ln^{2}(a)}{a^{4}} - \frac{12{a}^{b}ln(a)}{a^{4}} - \frac{6b{a}^{b}ln^{2}(a)}{a^{4}} + \frac{22{a}^{b}}{a^{4}} - \frac{36b{a}^{b}}{a^{4}} + \frac{12b^{2}{a}^{b}}{a^{4}} - sin(a)cos(b)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{8b^{3}{a}^{b}ln(a)}{a^{4}} + \frac{b^{4}{a}^{b}ln^{2}(a)}{a^{4}} - \frac{36b^{2}{a}^{b}ln(a)}{a^{4}} - \frac{6b^{3}{a}^{b}ln^{2}(a)}{a^{4}} + \frac{44b{a}^{b}ln(a)}{a^{4}} + \frac{11b^{2}{a}^{b}ln^{2}(a)}{a^{4}} - \frac{12{a}^{b}ln(a)}{a^{4}} - \frac{6b{a}^{b}ln^{2}(a)}{a^{4}} + \frac{22{a}^{b}}{a^{4}} - \frac{36b{a}^{b}}{a^{4}} + \frac{12b^{2}{a}^{b}}{a^{4}} - sin(a)cos(b)\right)}{db}\\=&\frac{8*3b^{2}{a}^{b}ln(a)}{a^{4}} + \frac{8b^{3}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln(a)}{a^{4}} + \frac{8b^{3}{a}^{b}*0}{a^{4}(a)} + \frac{4b^{3}{a}^{b}ln^{2}(a)}{a^{4}} + \frac{b^{4}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{2}(a)}{a^{4}} + \frac{b^{4}{a}^{b}*2ln(a)*0}{a^{4}(a)} - \frac{36*2b{a}^{b}ln(a)}{a^{4}} - \frac{36b^{2}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln(a)}{a^{4}} - \frac{36b^{2}{a}^{b}*0}{a^{4}(a)} - \frac{6*3b^{2}{a}^{b}ln^{2}(a)}{a^{4}} - \frac{6b^{3}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{2}(a)}{a^{4}} - \frac{6b^{3}{a}^{b}*2ln(a)*0}{a^{4}(a)} + \frac{44{a}^{b}ln(a)}{a^{4}} + \frac{44b({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln(a)}{a^{4}} + \frac{44b{a}^{b}*0}{a^{4}(a)} + \frac{11*2b{a}^{b}ln^{2}(a)}{a^{4}} + \frac{11b^{2}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{2}(a)}{a^{4}} + \frac{11b^{2}{a}^{b}*2ln(a)*0}{a^{4}(a)} - \frac{12({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln(a)}{a^{4}} - \frac{12{a}^{b}*0}{a^{4}(a)} - \frac{6{a}^{b}ln^{2}(a)}{a^{4}} - \frac{6b({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{2}(a)}{a^{4}} - \frac{6b{a}^{b}*2ln(a)*0}{a^{4}(a)} + \frac{22({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} - \frac{36{a}^{b}}{a^{4}} - \frac{36b({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} + \frac{12*2b{a}^{b}}{a^{4}} + \frac{12b^{2}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} - cos(a)*0cos(b) - sin(a)*-sin(b)\\=&\frac{36b^{2}{a}^{b}ln(a)}{a^{4}} + \frac{12b^{3}{a}^{b}ln^{2}(a)}{a^{4}} + \frac{b^{4}{a}^{b}ln^{3}(a)}{a^{4}} - \frac{108b{a}^{b}ln(a)}{a^{4}} - \frac{54b^{2}{a}^{b}ln^{2}(a)}{a^{4}} - \frac{6b^{3}{a}^{b}ln^{3}(a)}{a^{4}} + \frac{66{a}^{b}ln(a)}{a^{4}} + \frac{66b{a}^{b}ln^{2}(a)}{a^{4}} + \frac{11b^{2}{a}^{b}ln^{3}(a)}{a^{4}} - \frac{18{a}^{b}ln^{2}(a)}{a^{4}} - \frac{6b{a}^{b}ln^{3}(a)}{a^{4}} - \frac{36{a}^{b}}{a^{4}} + \frac{24b{a}^{b}}{a^{4}} + sin(b)sin(a)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{36b^{2}{a}^{b}ln(a)}{a^{4}} + \frac{12b^{3}{a}^{b}ln^{2}(a)}{a^{4}} + \frac{b^{4}{a}^{b}ln^{3}(a)}{a^{4}} - \frac{108b{a}^{b}ln(a)}{a^{4}} - \frac{54b^{2}{a}^{b}ln^{2}(a)}{a^{4}} - \frac{6b^{3}{a}^{b}ln^{3}(a)}{a^{4}} + \frac{66{a}^{b}ln(a)}{a^{4}} + \frac{66b{a}^{b}ln^{2}(a)}{a^{4}} + \frac{11b^{2}{a}^{b}ln^{3}(a)}{a^{4}} - \frac{18{a}^{b}ln^{2}(a)}{a^{4}} - \frac{6b{a}^{b}ln^{3}(a)}{a^{4}} - \frac{36{a}^{b}}{a^{4}} + \frac{24b{a}^{b}}{a^{4}} + sin(b)sin(a)\right)}{db}\\=&\frac{36*2b{a}^{b}ln(a)}{a^{4}} + \frac{36b^{2}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln(a)}{a^{4}} + \frac{36b^{2}{a}^{b}*0}{a^{4}(a)} + \frac{12*3b^{2}{a}^{b}ln^{2}(a)}{a^{4}} + \frac{12b^{3}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{2}(a)}{a^{4}} + \frac{12b^{3}{a}^{b}*2ln(a)*0}{a^{4}(a)} + \frac{4b^{3}{a}^{b}ln^{3}(a)}{a^{4}} + \frac{b^{4}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{3}(a)}{a^{4}} + \frac{b^{4}{a}^{b}*3ln^{2}(a)*0}{a^{4}(a)} - \frac{108{a}^{b}ln(a)}{a^{4}} - \frac{108b({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln(a)}{a^{4}} - \frac{108b{a}^{b}*0}{a^{4}(a)} - \frac{54*2b{a}^{b}ln^{2}(a)}{a^{4}} - \frac{54b^{2}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{2}(a)}{a^{4}} - \frac{54b^{2}{a}^{b}*2ln(a)*0}{a^{4}(a)} - \frac{6*3b^{2}{a}^{b}ln^{3}(a)}{a^{4}} - \frac{6b^{3}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{3}(a)}{a^{4}} - \frac{6b^{3}{a}^{b}*3ln^{2}(a)*0}{a^{4}(a)} + \frac{66({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln(a)}{a^{4}} + \frac{66{a}^{b}*0}{a^{4}(a)} + \frac{66{a}^{b}ln^{2}(a)}{a^{4}} + \frac{66b({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{2}(a)}{a^{4}} + \frac{66b{a}^{b}*2ln(a)*0}{a^{4}(a)} + \frac{11*2b{a}^{b}ln^{3}(a)}{a^{4}} + \frac{11b^{2}({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{3}(a)}{a^{4}} + \frac{11b^{2}{a}^{b}*3ln^{2}(a)*0}{a^{4}(a)} - \frac{18({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{2}(a)}{a^{4}} - \frac{18{a}^{b}*2ln(a)*0}{a^{4}(a)} - \frac{6{a}^{b}ln^{3}(a)}{a^{4}} - \frac{6b({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))ln^{3}(a)}{a^{4}} - \frac{6b{a}^{b}*3ln^{2}(a)*0}{a^{4}(a)} - \frac{36({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} + \frac{24{a}^{b}}{a^{4}} + \frac{24b({a}^{b}((1)ln(a) + \frac{(b)(0)}{(a)}))}{a^{4}} + cos(b)sin(a) + sin(b)cos(a)*0\\=&\frac{96b{a}^{b}ln(a)}{a^{4}} + \frac{72b^{2}{a}^{b}ln^{2}(a)}{a^{4}} + \frac{16b^{3}{a}^{b}ln^{3}(a)}{a^{4}} + \frac{b^{4}{a}^{b}ln^{4}(a)}{a^{4}} - \frac{144{a}^{b}ln(a)}{a^{4}} - \frac{216b{a}^{b}ln^{2}(a)}{a^{4}} - \frac{72b^{2}{a}^{b}ln^{3}(a)}{a^{4}} - \frac{6b^{3}{a}^{b}ln^{4}(a)}{a^{4}} + \frac{132{a}^{b}ln^{2}(a)}{a^{4}} + \frac{88b{a}^{b}ln^{3}(a)}{a^{4}} + \frac{11b^{2}{a}^{b}ln^{4}(a)}{a^{4}} - \frac{24{a}^{b}ln^{3}(a)}{a^{4}} - \frac{6b{a}^{b}ln^{4}(a)}{a^{4}} + \frac{24{a}^{b}}{a^{4}} + sin(a)cos(b)\\ \end{split}\end{equation} \]