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 i is calculated.
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {2}^{(a + bi)}(\frac{1}{(a + bi)} + {(a + bi)}^{3})\ with\ respect\ to\ i：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{{2}^{(a + bi)}}{(a + bi)} + a^{3}{2}^{(a + bi)} + 3a^{2}bi{2}^{(a + bi)} + 3ab^{2}i^{2}{2}^{(a + bi)} + b^{3}i^{3}{2}^{(a + bi)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{{2}^{(a + bi)}}{(a + bi)} + a^{3}{2}^{(a + bi)} + 3a^{2}bi{2}^{(a + bi)} + 3ab^{2}i^{2}{2}^{(a + bi)} + b^{3}i^{3}{2}^{(a + bi)}\right)}{di}\\=&(\frac{-(0 + b)}{(a + bi)^{2}}){2}^{(a + bi)} + \frac{({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))}{(a + bi)} + a^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)})) + 3a^{2}b{2}^{(a + bi)} + 3a^{2}bi({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)})) + 3ab^{2}*2i{2}^{(a + bi)} + 3ab^{2}i^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)})) + b^{3}*3i^{2}{2}^{(a + bi)} + b^{3}i^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))\\=&\frac{b{2}^{(a + bi)}ln(2)}{(a + bi)} - \frac{b{2}^{(a + bi)}}{(a + bi)^{2}} + a^{3}b{2}^{(a + bi)}ln(2) + 3a^{2}b{2}^{(a + bi)} + 3a^{2}b^{2}i{2}^{(a + bi)}ln(2) + 3ab^{3}i^{2}{2}^{(a + bi)}ln(2) + 6ab^{2}i{2}^{(a + bi)} + b^{4}i^{3}{2}^{(a + bi)}ln(2) + 3b^{3}i^{2}{2}^{(a + bi)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{b{2}^{(a + bi)}ln(2)}{(a + bi)} - \frac{b{2}^{(a + bi)}}{(a + bi)^{2}} + a^{3}b{2}^{(a + bi)}ln(2) + 3a^{2}b{2}^{(a + bi)} + 3a^{2}b^{2}i{2}^{(a + bi)}ln(2) + 3ab^{3}i^{2}{2}^{(a + bi)}ln(2) + 6ab^{2}i{2}^{(a + bi)} + b^{4}i^{3}{2}^{(a + bi)}ln(2) + 3b^{3}i^{2}{2}^{(a + bi)}\right)}{di}\\=&(\frac{-(0 + b)}{(a + bi)^{2}})b{2}^{(a + bi)}ln(2) + \frac{b({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2)}{(a + bi)} + \frac{b{2}^{(a + bi)}*0}{(a + bi)(2)} - (\frac{-2(0 + b)}{(a + bi)^{3}})b{2}^{(a + bi)} - \frac{b({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))}{(a + bi)^{2}} + a^{3}b({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2) + \frac{a^{3}b{2}^{(a + bi)}*0}{(2)} + 3a^{2}b({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)})) + 3a^{2}b^{2}{2}^{(a + bi)}ln(2) + 3a^{2}b^{2}i({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2) + \frac{3a^{2}b^{2}i{2}^{(a + bi)}*0}{(2)} + 3ab^{3}*2i{2}^{(a + bi)}ln(2) + 3ab^{3}i^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2) + \frac{3ab^{3}i^{2}{2}^{(a + bi)}*0}{(2)} + 6ab^{2}{2}^{(a + bi)} + 6ab^{2}i({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)})) + b^{4}*3i^{2}{2}^{(a + bi)}ln(2) + b^{4}i^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2) + \frac{b^{4}i^{3}{2}^{(a + bi)}*0}{(2)} + 3b^{3}*2i{2}^{(a + bi)} + 3b^{3}i^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))\\=& - \frac{2b^{2}{2}^{(a + bi)}ln(2)}{(a + bi)^{2}} + \frac{b^{2}{2}^{(a + bi)}ln^{2}(2)}{(a + bi)} + \frac{2b^{2}{2}^{(a + bi)}}{(a + bi)^{3}} + a^{3}b^{2}{2}^{(a + bi)}ln^{2}(2) + 6a^{2}b^{2}{2}^{(a + bi)}ln(2) + 3a^{2}b^{3}i{2}^{(a + bi)}ln^{2}(2) + 12ab^{3}i{2}^{(a + bi)}ln(2) + 3ab^{4}i^{2}{2}^{(a + bi)}ln^{2}(2) + 6ab^{2}{2}^{(a + bi)} + 6b^{4}i^{2}{2}^{(a + bi)}ln(2) + b^{5}i^{3}{2}^{(a + bi)}ln^{2}(2) + 6b^{3}i{2}^{(a + bi)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( - \frac{2b^{2}{2}^{(a + bi)}ln(2)}{(a + bi)^{2}} + \frac{b^{2}{2}^{(a + bi)}ln^{2}(2)}{(a + bi)} + \frac{2b^{2}{2}^{(a + bi)}}{(a + bi)^{3}} + a^{3}b^{2}{2}^{(a + bi)}ln^{2}(2) + 6a^{2}b^{2}{2}^{(a + bi)}ln(2) + 3a^{2}b^{3}i{2}^{(a + bi)}ln^{2}(2) + 12ab^{3}i{2}^{(a + bi)}ln(2) + 3ab^{4}i^{2}{2}^{(a + bi)}ln^{2}(2) + 6ab^{2}{2}^{(a + bi)} + 6b^{4}i^{2}{2}^{(a + bi)}ln(2) + b^{5}i^{3}{2}^{(a + bi)}ln^{2}(2) + 6b^{3}i{2}^{(a + bi)}\right)}{di}\\=& - 2(\frac{-2(0 + b)}{(a + bi)^{3}})b^{2}{2}^{(a + bi)}ln(2) - \frac{2b^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2)}{(a + bi)^{2}} - \frac{2b^{2}{2}^{(a + bi)}*0}{(a + bi)^{2}(2)} + (\frac{-(0 + b)}{(a + bi)^{2}})b^{2}{2}^{(a + bi)}ln^{2}(2) + \frac{b^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{2}(2)}{(a + bi)} + \frac{b^{2}{2}^{(a + bi)}*2ln(2)*0}{(a + bi)(2)} + 2(\frac{-3(0 + b)}{(a + bi)^{4}})b^{2}{2}^{(a + bi)} + \frac{2b^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))}{(a + bi)^{3}} + a^{3}b^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{2}(2) + \frac{a^{3}b^{2}{2}^{(a + bi)}*2ln(2)*0}{(2)} + 6a^{2}b^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2) + \frac{6a^{2}b^{2}{2}^{(a + bi)}*0}{(2)} + 3a^{2}b^{3}{2}^{(a + bi)}ln^{2}(2) + 3a^{2}b^{3}i({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{2}(2) + \frac{3a^{2}b^{3}i{2}^{(a + bi)}*2ln(2)*0}{(2)} + 12ab^{3}{2}^{(a + bi)}ln(2) + 12ab^{3}i({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2) + \frac{12ab^{3}i{2}^{(a + bi)}*0}{(2)} + 3ab^{4}*2i{2}^{(a + bi)}ln^{2}(2) + 3ab^{4}i^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{2}(2) + \frac{3ab^{4}i^{2}{2}^{(a + bi)}*2ln(2)*0}{(2)} + 6ab^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)})) + 6b^{4}*2i{2}^{(a + bi)}ln(2) + 6b^{4}i^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2) + \frac{6b^{4}i^{2}{2}^{(a + bi)}*0}{(2)} + b^{5}*3i^{2}{2}^{(a + bi)}ln^{2}(2) + b^{5}i^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{2}(2) + \frac{b^{5}i^{3}{2}^{(a + bi)}*2ln(2)*0}{(2)} + 6b^{3}{2}^{(a + bi)} + 6b^{3}i({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))\\=&\frac{6b^{3}{2}^{(a + bi)}ln(2)}{(a + bi)^{3}} - \frac{3b^{3}{2}^{(a + bi)}ln^{2}(2)}{(a + bi)^{2}} + \frac{b^{3}{2}^{(a + bi)}ln^{3}(2)}{(a + bi)} - \frac{6b^{3}{2}^{(a + bi)}}{(a + bi)^{4}} + a^{3}b^{3}{2}^{(a + bi)}ln^{3}(2) + 9a^{2}b^{3}{2}^{(a + bi)}ln^{2}(2) + 3a^{2}b^{4}i{2}^{(a + bi)}ln^{3}(2) + 18ab^{3}{2}^{(a + bi)}ln(2) + 18ab^{4}i{2}^{(a + bi)}ln^{2}(2) + 3ab^{5}i^{2}{2}^{(a + bi)}ln^{3}(2) + 18b^{4}i{2}^{(a + bi)}ln(2) + 9b^{5}i^{2}{2}^{(a + bi)}ln^{2}(2) + b^{6}i^{3}{2}^{(a + bi)}ln^{3}(2) + 6b^{3}{2}^{(a + bi)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{6b^{3}{2}^{(a + bi)}ln(2)}{(a + bi)^{3}} - \frac{3b^{3}{2}^{(a + bi)}ln^{2}(2)}{(a + bi)^{2}} + \frac{b^{3}{2}^{(a + bi)}ln^{3}(2)}{(a + bi)} - \frac{6b^{3}{2}^{(a + bi)}}{(a + bi)^{4}} + a^{3}b^{3}{2}^{(a + bi)}ln^{3}(2) + 9a^{2}b^{3}{2}^{(a + bi)}ln^{2}(2) + 3a^{2}b^{4}i{2}^{(a + bi)}ln^{3}(2) + 18ab^{3}{2}^{(a + bi)}ln(2) + 18ab^{4}i{2}^{(a + bi)}ln^{2}(2) + 3ab^{5}i^{2}{2}^{(a + bi)}ln^{3}(2) + 18b^{4}i{2}^{(a + bi)}ln(2) + 9b^{5}i^{2}{2}^{(a + bi)}ln^{2}(2) + b^{6}i^{3}{2}^{(a + bi)}ln^{3}(2) + 6b^{3}{2}^{(a + bi)}\right)}{di}\\=&6(\frac{-3(0 + b)}{(a + bi)^{4}})b^{3}{2}^{(a + bi)}ln(2) + \frac{6b^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2)}{(a + bi)^{3}} + \frac{6b^{3}{2}^{(a + bi)}*0}{(a + bi)^{3}(2)} - 3(\frac{-2(0 + b)}{(a + bi)^{3}})b^{3}{2}^{(a + bi)}ln^{2}(2) - \frac{3b^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{2}(2)}{(a + bi)^{2}} - \frac{3b^{3}{2}^{(a + bi)}*2ln(2)*0}{(a + bi)^{2}(2)} + (\frac{-(0 + b)}{(a + bi)^{2}})b^{3}{2}^{(a + bi)}ln^{3}(2) + \frac{b^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{3}(2)}{(a + bi)} + \frac{b^{3}{2}^{(a + bi)}*3ln^{2}(2)*0}{(a + bi)(2)} - 6(\frac{-4(0 + b)}{(a + bi)^{5}})b^{3}{2}^{(a + bi)} - \frac{6b^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))}{(a + bi)^{4}} + a^{3}b^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{3}(2) + \frac{a^{3}b^{3}{2}^{(a + bi)}*3ln^{2}(2)*0}{(2)} + 9a^{2}b^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{2}(2) + \frac{9a^{2}b^{3}{2}^{(a + bi)}*2ln(2)*0}{(2)} + 3a^{2}b^{4}{2}^{(a + bi)}ln^{3}(2) + 3a^{2}b^{4}i({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{3}(2) + \frac{3a^{2}b^{4}i{2}^{(a + bi)}*3ln^{2}(2)*0}{(2)} + 18ab^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2) + \frac{18ab^{3}{2}^{(a + bi)}*0}{(2)} + 18ab^{4}{2}^{(a + bi)}ln^{2}(2) + 18ab^{4}i({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{2}(2) + \frac{18ab^{4}i{2}^{(a + bi)}*2ln(2)*0}{(2)} + 3ab^{5}*2i{2}^{(a + bi)}ln^{3}(2) + 3ab^{5}i^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{3}(2) + \frac{3ab^{5}i^{2}{2}^{(a + bi)}*3ln^{2}(2)*0}{(2)} + 18b^{4}{2}^{(a + bi)}ln(2) + 18b^{4}i({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2) + \frac{18b^{4}i{2}^{(a + bi)}*0}{(2)} + 9b^{5}*2i{2}^{(a + bi)}ln^{2}(2) + 9b^{5}i^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{2}(2) + \frac{9b^{5}i^{2}{2}^{(a + bi)}*2ln(2)*0}{(2)} + b^{6}*3i^{2}{2}^{(a + bi)}ln^{3}(2) + b^{6}i^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{3}(2) + \frac{b^{6}i^{3}{2}^{(a + bi)}*3ln^{2}(2)*0}{(2)} + 6b^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))\\=& - \frac{24b^{4}{2}^{(a + bi)}ln(2)}{(a + bi)^{4}} + \frac{12b^{4}{2}^{(a + bi)}ln^{2}(2)}{(a + bi)^{3}} - \frac{4b^{4}{2}^{(a + bi)}ln^{3}(2)}{(a + bi)^{2}} + \frac{b^{4}{2}^{(a + bi)}ln^{4}(2)}{(a + bi)} + \frac{24b^{4}{2}^{(a + bi)}}{(a + bi)^{5}} + a^{3}b^{4}{2}^{(a + bi)}ln^{4}(2) + 12a^{2}b^{4}{2}^{(a + bi)}ln^{3}(2) + 3a^{2}b^{5}i{2}^{(a + bi)}ln^{4}(2) + 36ab^{4}{2}^{(a + bi)}ln^{2}(2) + 24ab^{5}i{2}^{(a + bi)}ln^{3}(2) + 3ab^{6}i^{2}{2}^{(a + bi)}ln^{4}(2) + 24b^{4}{2}^{(a + bi)}ln(2) + 36b^{5}i{2}^{(a + bi)}ln^{2}(2) + 12b^{6}i^{2}{2}^{(a + bi)}ln^{3}(2) + b^{7}i^{3}{2}^{(a + bi)}ln^{4}(2)\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {2}^{(a + bi)}(\frac{1}{(a + bi)} + {(a + bi)}^{3})\ with\ respect\ to\ i：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{{2}^{(a + bi)}}{(a + bi)} + a^{3}{2}^{(a + bi)} + 3a^{2}bi{2}^{(a + bi)} + 3ab^{2}i^{2}{2}^{(a + bi)} + b^{3}i^{3}{2}^{(a + bi)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{{2}^{(a + bi)}}{(a + bi)} + a^{3}{2}^{(a + bi)} + 3a^{2}bi{2}^{(a + bi)} + 3ab^{2}i^{2}{2}^{(a + bi)} + b^{3}i^{3}{2}^{(a + bi)}\right)}{di}\\=&(\frac{-(0 + b)}{(a + bi)^{2}}){2}^{(a + bi)} + \frac{({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))}{(a + bi)} + a^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)})) + 3a^{2}b{2}^{(a + bi)} + 3a^{2}bi({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)})) + 3ab^{2}*2i{2}^{(a + bi)} + 3ab^{2}i^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)})) + b^{3}*3i^{2}{2}^{(a + bi)} + b^{3}i^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))\\=&\frac{b{2}^{(a + bi)}ln(2)}{(a + bi)} - \frac{b{2}^{(a + bi)}}{(a + bi)^{2}} + a^{3}b{2}^{(a + bi)}ln(2) + 3a^{2}b{2}^{(a + bi)} + 3a^{2}b^{2}i{2}^{(a + bi)}ln(2) + 3ab^{3}i^{2}{2}^{(a + bi)}ln(2) + 6ab^{2}i{2}^{(a + bi)} + b^{4}i^{3}{2}^{(a + bi)}ln(2) + 3b^{3}i^{2}{2}^{(a + bi)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{b{2}^{(a + bi)}ln(2)}{(a + bi)} - \frac{b{2}^{(a + bi)}}{(a + bi)^{2}} + a^{3}b{2}^{(a + bi)}ln(2) + 3a^{2}b{2}^{(a + bi)} + 3a^{2}b^{2}i{2}^{(a + bi)}ln(2) + 3ab^{3}i^{2}{2}^{(a + bi)}ln(2) + 6ab^{2}i{2}^{(a + bi)} + b^{4}i^{3}{2}^{(a + bi)}ln(2) + 3b^{3}i^{2}{2}^{(a + bi)}\right)}{di}\\=&(\frac{-(0 + b)}{(a + bi)^{2}})b{2}^{(a + bi)}ln(2) + \frac{b({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2)}{(a + bi)} + \frac{b{2}^{(a + bi)}*0}{(a + bi)(2)} - (\frac{-2(0 + b)}{(a + bi)^{3}})b{2}^{(a + bi)} - \frac{b({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))}{(a + bi)^{2}} + a^{3}b({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2) + \frac{a^{3}b{2}^{(a + bi)}*0}{(2)} + 3a^{2}b({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)})) + 3a^{2}b^{2}{2}^{(a + bi)}ln(2) + 3a^{2}b^{2}i({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2) + \frac{3a^{2}b^{2}i{2}^{(a + bi)}*0}{(2)} + 3ab^{3}*2i{2}^{(a + bi)}ln(2) + 3ab^{3}i^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2) + \frac{3ab^{3}i^{2}{2}^{(a + bi)}*0}{(2)} + 6ab^{2}{2}^{(a + bi)} + 6ab^{2}i({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)})) + b^{4}*3i^{2}{2}^{(a + bi)}ln(2) + b^{4}i^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2) + \frac{b^{4}i^{3}{2}^{(a + bi)}*0}{(2)} + 3b^{3}*2i{2}^{(a + bi)} + 3b^{3}i^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))\\=& - \frac{2b^{2}{2}^{(a + bi)}ln(2)}{(a + bi)^{2}} + \frac{b^{2}{2}^{(a + bi)}ln^{2}(2)}{(a + bi)} + \frac{2b^{2}{2}^{(a + bi)}}{(a + bi)^{3}} + a^{3}b^{2}{2}^{(a + bi)}ln^{2}(2) + 6a^{2}b^{2}{2}^{(a + bi)}ln(2) + 3a^{2}b^{3}i{2}^{(a + bi)}ln^{2}(2) + 12ab^{3}i{2}^{(a + bi)}ln(2) + 3ab^{4}i^{2}{2}^{(a + bi)}ln^{2}(2) + 6ab^{2}{2}^{(a + bi)} + 6b^{4}i^{2}{2}^{(a + bi)}ln(2) + b^{5}i^{3}{2}^{(a + bi)}ln^{2}(2) + 6b^{3}i{2}^{(a + bi)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( - \frac{2b^{2}{2}^{(a + bi)}ln(2)}{(a + bi)^{2}} + \frac{b^{2}{2}^{(a + bi)}ln^{2}(2)}{(a + bi)} + \frac{2b^{2}{2}^{(a + bi)}}{(a + bi)^{3}} + a^{3}b^{2}{2}^{(a + bi)}ln^{2}(2) + 6a^{2}b^{2}{2}^{(a + bi)}ln(2) + 3a^{2}b^{3}i{2}^{(a + bi)}ln^{2}(2) + 12ab^{3}i{2}^{(a + bi)}ln(2) + 3ab^{4}i^{2}{2}^{(a + bi)}ln^{2}(2) + 6ab^{2}{2}^{(a + bi)} + 6b^{4}i^{2}{2}^{(a + bi)}ln(2) + b^{5}i^{3}{2}^{(a + bi)}ln^{2}(2) + 6b^{3}i{2}^{(a + bi)}\right)}{di}\\=& - 2(\frac{-2(0 + b)}{(a + bi)^{3}})b^{2}{2}^{(a + bi)}ln(2) - \frac{2b^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2)}{(a + bi)^{2}} - \frac{2b^{2}{2}^{(a + bi)}*0}{(a + bi)^{2}(2)} + (\frac{-(0 + b)}{(a + bi)^{2}})b^{2}{2}^{(a + bi)}ln^{2}(2) + \frac{b^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{2}(2)}{(a + bi)} + \frac{b^{2}{2}^{(a + bi)}*2ln(2)*0}{(a + bi)(2)} + 2(\frac{-3(0 + b)}{(a + bi)^{4}})b^{2}{2}^{(a + bi)} + \frac{2b^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))}{(a + bi)^{3}} + a^{3}b^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{2}(2) + \frac{a^{3}b^{2}{2}^{(a + bi)}*2ln(2)*0}{(2)} + 6a^{2}b^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2) + \frac{6a^{2}b^{2}{2}^{(a + bi)}*0}{(2)} + 3a^{2}b^{3}{2}^{(a + bi)}ln^{2}(2) + 3a^{2}b^{3}i({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{2}(2) + \frac{3a^{2}b^{3}i{2}^{(a + bi)}*2ln(2)*0}{(2)} + 12ab^{3}{2}^{(a + bi)}ln(2) + 12ab^{3}i({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2) + \frac{12ab^{3}i{2}^{(a + bi)}*0}{(2)} + 3ab^{4}*2i{2}^{(a + bi)}ln^{2}(2) + 3ab^{4}i^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{2}(2) + \frac{3ab^{4}i^{2}{2}^{(a + bi)}*2ln(2)*0}{(2)} + 6ab^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)})) + 6b^{4}*2i{2}^{(a + bi)}ln(2) + 6b^{4}i^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2) + \frac{6b^{4}i^{2}{2}^{(a + bi)}*0}{(2)} + b^{5}*3i^{2}{2}^{(a + bi)}ln^{2}(2) + b^{5}i^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{2}(2) + \frac{b^{5}i^{3}{2}^{(a + bi)}*2ln(2)*0}{(2)} + 6b^{3}{2}^{(a + bi)} + 6b^{3}i({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))\\=&\frac{6b^{3}{2}^{(a + bi)}ln(2)}{(a + bi)^{3}} - \frac{3b^{3}{2}^{(a + bi)}ln^{2}(2)}{(a + bi)^{2}} + \frac{b^{3}{2}^{(a + bi)}ln^{3}(2)}{(a + bi)} - \frac{6b^{3}{2}^{(a + bi)}}{(a + bi)^{4}} + a^{3}b^{3}{2}^{(a + bi)}ln^{3}(2) + 9a^{2}b^{3}{2}^{(a + bi)}ln^{2}(2) + 3a^{2}b^{4}i{2}^{(a + bi)}ln^{3}(2) + 18ab^{3}{2}^{(a + bi)}ln(2) + 18ab^{4}i{2}^{(a + bi)}ln^{2}(2) + 3ab^{5}i^{2}{2}^{(a + bi)}ln^{3}(2) + 18b^{4}i{2}^{(a + bi)}ln(2) + 9b^{5}i^{2}{2}^{(a + bi)}ln^{2}(2) + b^{6}i^{3}{2}^{(a + bi)}ln^{3}(2) + 6b^{3}{2}^{(a + bi)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{6b^{3}{2}^{(a + bi)}ln(2)}{(a + bi)^{3}} - \frac{3b^{3}{2}^{(a + bi)}ln^{2}(2)}{(a + bi)^{2}} + \frac{b^{3}{2}^{(a + bi)}ln^{3}(2)}{(a + bi)} - \frac{6b^{3}{2}^{(a + bi)}}{(a + bi)^{4}} + a^{3}b^{3}{2}^{(a + bi)}ln^{3}(2) + 9a^{2}b^{3}{2}^{(a + bi)}ln^{2}(2) + 3a^{2}b^{4}i{2}^{(a + bi)}ln^{3}(2) + 18ab^{3}{2}^{(a + bi)}ln(2) + 18ab^{4}i{2}^{(a + bi)}ln^{2}(2) + 3ab^{5}i^{2}{2}^{(a + bi)}ln^{3}(2) + 18b^{4}i{2}^{(a + bi)}ln(2) + 9b^{5}i^{2}{2}^{(a + bi)}ln^{2}(2) + b^{6}i^{3}{2}^{(a + bi)}ln^{3}(2) + 6b^{3}{2}^{(a + bi)}\right)}{di}\\=&6(\frac{-3(0 + b)}{(a + bi)^{4}})b^{3}{2}^{(a + bi)}ln(2) + \frac{6b^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2)}{(a + bi)^{3}} + \frac{6b^{3}{2}^{(a + bi)}*0}{(a + bi)^{3}(2)} - 3(\frac{-2(0 + b)}{(a + bi)^{3}})b^{3}{2}^{(a + bi)}ln^{2}(2) - \frac{3b^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{2}(2)}{(a + bi)^{2}} - \frac{3b^{3}{2}^{(a + bi)}*2ln(2)*0}{(a + bi)^{2}(2)} + (\frac{-(0 + b)}{(a + bi)^{2}})b^{3}{2}^{(a + bi)}ln^{3}(2) + \frac{b^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{3}(2)}{(a + bi)} + \frac{b^{3}{2}^{(a + bi)}*3ln^{2}(2)*0}{(a + bi)(2)} - 6(\frac{-4(0 + b)}{(a + bi)^{5}})b^{3}{2}^{(a + bi)} - \frac{6b^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))}{(a + bi)^{4}} + a^{3}b^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{3}(2) + \frac{a^{3}b^{3}{2}^{(a + bi)}*3ln^{2}(2)*0}{(2)} + 9a^{2}b^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{2}(2) + \frac{9a^{2}b^{3}{2}^{(a + bi)}*2ln(2)*0}{(2)} + 3a^{2}b^{4}{2}^{(a + bi)}ln^{3}(2) + 3a^{2}b^{4}i({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{3}(2) + \frac{3a^{2}b^{4}i{2}^{(a + bi)}*3ln^{2}(2)*0}{(2)} + 18ab^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2) + \frac{18ab^{3}{2}^{(a + bi)}*0}{(2)} + 18ab^{4}{2}^{(a + bi)}ln^{2}(2) + 18ab^{4}i({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{2}(2) + \frac{18ab^{4}i{2}^{(a + bi)}*2ln(2)*0}{(2)} + 3ab^{5}*2i{2}^{(a + bi)}ln^{3}(2) + 3ab^{5}i^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{3}(2) + \frac{3ab^{5}i^{2}{2}^{(a + bi)}*3ln^{2}(2)*0}{(2)} + 18b^{4}{2}^{(a + bi)}ln(2) + 18b^{4}i({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln(2) + \frac{18b^{4}i{2}^{(a + bi)}*0}{(2)} + 9b^{5}*2i{2}^{(a + bi)}ln^{2}(2) + 9b^{5}i^{2}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{2}(2) + \frac{9b^{5}i^{2}{2}^{(a + bi)}*2ln(2)*0}{(2)} + b^{6}*3i^{2}{2}^{(a + bi)}ln^{3}(2) + b^{6}i^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))ln^{3}(2) + \frac{b^{6}i^{3}{2}^{(a + bi)}*3ln^{2}(2)*0}{(2)} + 6b^{3}({2}^{(a + bi)}((0 + b)ln(2) + \frac{(a + bi)(0)}{(2)}))\\=& - \frac{24b^{4}{2}^{(a + bi)}ln(2)}{(a + bi)^{4}} + \frac{12b^{4}{2}^{(a + bi)}ln^{2}(2)}{(a + bi)^{3}} - \frac{4b^{4}{2}^{(a + bi)}ln^{3}(2)}{(a + bi)^{2}} + \frac{b^{4}{2}^{(a + bi)}ln^{4}(2)}{(a + bi)} + \frac{24b^{4}{2}^{(a + bi)}}{(a + bi)^{5}} + a^{3}b^{4}{2}^{(a + bi)}ln^{4}(2) + 12a^{2}b^{4}{2}^{(a + bi)}ln^{3}(2) + 3a^{2}b^{5}i{2}^{(a + bi)}ln^{4}(2) + 36ab^{4}{2}^{(a + bi)}ln^{2}(2) + 24ab^{5}i{2}^{(a + bi)}ln^{3}(2) + 3ab^{6}i^{2}{2}^{(a + bi)}ln^{4}(2) + 24b^{4}{2}^{(a + bi)}ln(2) + 36b^{5}i{2}^{(a + bi)}ln^{2}(2) + 12b^{6}i^{2}{2}^{(a + bi)}ln^{3}(2) + b^{7}i^{3}{2}^{(a + bi)}ln^{4}(2)\\ \end{split}\end{equation} \]