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.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {ln(xx + x + 7)}^{\frac{1}{2}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = ln^{\frac{1}{2}}(x^{2} + x + 7)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( ln^{\frac{1}{2}}(x^{2} + x + 7)\right)}{dx}\\=&\frac{\frac{1}{2}(2x + 1 + 0)}{ln^{\frac{1}{2}}(x^{2} + x + 7)(x^{2} + x + 7)}\\=&\frac{x}{(x^{2} + x + 7)ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{1}{2(x^{2} + x + 7)ln^{\frac{1}{2}}(x^{2} + x + 7)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{x}{(x^{2} + x + 7)ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{1}{2(x^{2} + x + 7)ln^{\frac{1}{2}}(x^{2} + x + 7)}\right)}{dx}\\=&\frac{(\frac{-(2x + 1 + 0)}{(x^{2} + x + 7)^{2}})x}{ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{1}{(x^{2} + x + 7)ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{x*\frac{-1}{2}(2x + 1 + 0)}{(x^{2} + x + 7)ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{(\frac{-(2x + 1 + 0)}{(x^{2} + x + 7)^{2}})}{2ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{\frac{-1}{2}(2x + 1 + 0)}{2(x^{2} + x + 7)ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)}\\=&\frac{-2x^{2}}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{2x}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{1}{(x^{2} + x + 7)ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{x^{2}}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{x}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{1}{2(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{1}{4(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2x^{2}}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{2x}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{1}{(x^{2} + x + 7)ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{x^{2}}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{x}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{1}{2(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{1}{4(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)}\right)}{dx}\\=&\frac{-2(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})x^{2}}{ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{2*2x}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{2x^{2}*\frac{-1}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} - \frac{2(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})x}{ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{2}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{2x*\frac{-1}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{(\frac{-(2x + 1 + 0)}{(x^{2} + x + 7)^{2}})}{ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{\frac{-1}{2}(2x + 1 + 0)}{(x^{2} + x + 7)ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} - \frac{(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})x^{2}}{ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{2x}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{x^{2}*\frac{-3}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{2}ln^{\frac{5}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} - \frac{(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})x}{ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{1}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{x*\frac{-3}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{2}ln^{\frac{5}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} - \frac{(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})}{2ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{\frac{-1}{2}(2x + 1 + 0)}{2(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} - \frac{(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})}{4ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{\frac{-3}{2}(2x + 1 + 0)}{4(x^{2} + x + 7)^{2}ln^{\frac{5}{2}}(x^{2} + x + 7)(x^{2} + x + 7)}\\=&\frac{8x^{3}}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{12x^{2}}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{6x}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{6x^{3}}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{9x^{2}}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{6x}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{3}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{9x}{2(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{3x}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{3}{2(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{3x^{3}}{(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{9x^{2}}{2(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{9x}{4(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{1}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{3}{4(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{3}{8(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{8x^{3}}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{12x^{2}}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{6x}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{6x^{3}}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{9x^{2}}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{6x}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{3}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{9x}{2(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{3x}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{3}{2(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{3x^{3}}{(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{9x^{2}}{2(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{9x}{4(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{1}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{3}{4(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{3}{8(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)}\right)}{dx}\\=&\frac{8(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})x^{3}}{ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{8*3x^{2}}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{8x^{3}*\frac{-1}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{12(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})x^{2}}{ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{12*2x}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{12x^{2}*\frac{-1}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} - \frac{6(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})x}{ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{6}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{6x*\frac{-1}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{6(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})x^{3}}{ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{6*3x^{2}}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{6x^{3}*\frac{-3}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{9(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})x^{2}}{ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{9*2x}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{9x^{2}*\frac{-3}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{6(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})x}{ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{6}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{6x*\frac{-1}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} - \frac{3(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})}{ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{3*\frac{-1}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{9(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})x}{2ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{9}{2(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{9x*\frac{-3}{2}(2x + 1 + 0)}{2(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} - \frac{3(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})x}{ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{3}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{3x*\frac{-3}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{2}ln^{\frac{5}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} - \frac{3(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})}{2ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{3*\frac{-3}{2}(2x + 1 + 0)}{2(x^{2} + x + 7)^{2}ln^{\frac{5}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{3(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})x^{3}}{ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{3*3x^{2}}{(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{3x^{3}*\frac{-5}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{3}ln^{\frac{7}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{9(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})x^{2}}{2ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{9*2x}{2(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{9x^{2}*\frac{-5}{2}(2x + 1 + 0)}{2(x^{2} + x + 7)^{3}ln^{\frac{7}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{9(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})x}{4ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{9}{4(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{9x*\frac{-5}{2}(2x + 1 + 0)}{4(x^{2} + x + 7)^{3}ln^{\frac{7}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})}{ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{\frac{-1}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{3(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})}{4ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{3*\frac{-3}{2}(2x + 1 + 0)}{4(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{3(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})}{8ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{3*\frac{-5}{2}(2x + 1 + 0)}{8(x^{2} + x + 7)^{3}ln^{\frac{7}{2}}(x^{2} + x + 7)(x^{2} + x + 7)}\\=&\frac{-48x^{4}}{(x^{2} + x + 7)^{4}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{96x^{3}}{(x^{2} + x + 7)^{4}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{48x^{2}}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{44x^{4}}{(x^{2} + x + 7)^{4}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{88x^{3}}{(x^{2} + x + 7)^{4}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{72x^{2}}{(x^{2} + x + 7)^{4}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{48x}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{66x^{2}}{(x^{2} + x + 7)^{4}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{6}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{36x^{2}}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{36x}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{36x^{4}}{(x^{2} + x + 7)^{4}ln^{\frac{5}{2}}(x^{2} + x + 7)} - \frac{72x^{3}}{(x^{2} + x + 7)^{4}ln^{\frac{5}{2}}(x^{2} + x + 7)} - \frac{54x^{2}}{(x^{2} + x + 7)^{4}ln^{\frac{5}{2}}(x^{2} + x + 7)} - \frac{24x}{(x^{2} + x + 7)^{4}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{12}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{22x}{(x^{2} + x + 7)^{4}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{9}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{18x}{(x^{2} + x + 7)^{4}ln^{\frac{5}{2}}(x^{2} + x + 7)} - \frac{3}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{18x^{2}}{(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{18x}{(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{9}{2(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} - \frac{15x^{4}}{(x^{2} + x + 7)^{4}ln^{\frac{7}{2}}(x^{2} + x + 7)} - \frac{30x^{3}}{(x^{2} + x + 7)^{4}ln^{\frac{7}{2}}(x^{2} + x + 7)} - \frac{45x^{2}}{2(x^{2} + x + 7)^{4}ln^{\frac{7}{2}}(x^{2} + x + 7)} - \frac{15x}{2(x^{2} + x + 7)^{4}ln^{\frac{7}{2}}(x^{2} + x + 7)} - \frac{3}{(x^{2} + x + 7)^{4}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{11}{4(x^{2} + x + 7)^{4}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{9}{4(x^{2} + x + 7)^{4}ln^{\frac{5}{2}}(x^{2} + x + 7)} - \frac{15}{16(x^{2} + x + 7)^{4}ln^{\frac{7}{2}}(x^{2} + x + 7)}\\ \end{split}\end{equation} \]
Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {ln(xx + x + 7)}^{\frac{1}{2}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = ln^{\frac{1}{2}}(x^{2} + x + 7)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( ln^{\frac{1}{2}}(x^{2} + x + 7)\right)}{dx}\\=&\frac{\frac{1}{2}(2x + 1 + 0)}{ln^{\frac{1}{2}}(x^{2} + x + 7)(x^{2} + x + 7)}\\=&\frac{x}{(x^{2} + x + 7)ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{1}{2(x^{2} + x + 7)ln^{\frac{1}{2}}(x^{2} + x + 7)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{x}{(x^{2} + x + 7)ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{1}{2(x^{2} + x + 7)ln^{\frac{1}{2}}(x^{2} + x + 7)}\right)}{dx}\\=&\frac{(\frac{-(2x + 1 + 0)}{(x^{2} + x + 7)^{2}})x}{ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{1}{(x^{2} + x + 7)ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{x*\frac{-1}{2}(2x + 1 + 0)}{(x^{2} + x + 7)ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{(\frac{-(2x + 1 + 0)}{(x^{2} + x + 7)^{2}})}{2ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{\frac{-1}{2}(2x + 1 + 0)}{2(x^{2} + x + 7)ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)}\\=&\frac{-2x^{2}}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{2x}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{1}{(x^{2} + x + 7)ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{x^{2}}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{x}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{1}{2(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{1}{4(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2x^{2}}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{2x}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{1}{(x^{2} + x + 7)ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{x^{2}}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{x}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{1}{2(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{1}{4(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)}\right)}{dx}\\=&\frac{-2(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})x^{2}}{ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{2*2x}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{2x^{2}*\frac{-1}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} - \frac{2(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})x}{ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{2}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{2x*\frac{-1}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{(\frac{-(2x + 1 + 0)}{(x^{2} + x + 7)^{2}})}{ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{\frac{-1}{2}(2x + 1 + 0)}{(x^{2} + x + 7)ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} - \frac{(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})x^{2}}{ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{2x}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{x^{2}*\frac{-3}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{2}ln^{\frac{5}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} - \frac{(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})x}{ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{1}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{x*\frac{-3}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{2}ln^{\frac{5}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} - \frac{(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})}{2ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{\frac{-1}{2}(2x + 1 + 0)}{2(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} - \frac{(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})}{4ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{\frac{-3}{2}(2x + 1 + 0)}{4(x^{2} + x + 7)^{2}ln^{\frac{5}{2}}(x^{2} + x + 7)(x^{2} + x + 7)}\\=&\frac{8x^{3}}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{12x^{2}}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{6x}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{6x^{3}}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{9x^{2}}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{6x}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{3}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{9x}{2(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{3x}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{3}{2(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{3x^{3}}{(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{9x^{2}}{2(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{9x}{4(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{1}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{3}{4(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{3}{8(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{8x^{3}}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{12x^{2}}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{6x}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{6x^{3}}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{9x^{2}}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{6x}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{3}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{9x}{2(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{3x}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{3}{2(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{3x^{3}}{(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{9x^{2}}{2(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{9x}{4(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{1}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{3}{4(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{3}{8(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)}\right)}{dx}\\=&\frac{8(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})x^{3}}{ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{8*3x^{2}}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{8x^{3}*\frac{-1}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{12(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})x^{2}}{ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{12*2x}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{12x^{2}*\frac{-1}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} - \frac{6(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})x}{ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{6}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{6x*\frac{-1}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{6(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})x^{3}}{ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{6*3x^{2}}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{6x^{3}*\frac{-3}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{9(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})x^{2}}{ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{9*2x}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{9x^{2}*\frac{-3}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{6(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})x}{ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{6}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{6x*\frac{-1}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} - \frac{3(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})}{ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{3*\frac{-1}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{9(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})x}{2ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{9}{2(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{9x*\frac{-3}{2}(2x + 1 + 0)}{2(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} - \frac{3(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})x}{ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{3}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{3x*\frac{-3}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{2}ln^{\frac{5}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} - \frac{3(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})}{2ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{3*\frac{-3}{2}(2x + 1 + 0)}{2(x^{2} + x + 7)^{2}ln^{\frac{5}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{3(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})x^{3}}{ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{3*3x^{2}}{(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{3x^{3}*\frac{-5}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{3}ln^{\frac{7}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{9(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})x^{2}}{2ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{9*2x}{2(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{9x^{2}*\frac{-5}{2}(2x + 1 + 0)}{2(x^{2} + x + 7)^{3}ln^{\frac{7}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{9(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})x}{4ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{9}{4(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{9x*\frac{-5}{2}(2x + 1 + 0)}{4(x^{2} + x + 7)^{3}ln^{\frac{7}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})}{ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{\frac{-1}{2}(2x + 1 + 0)}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{3(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})}{4ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{3*\frac{-3}{2}(2x + 1 + 0)}{4(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)(x^{2} + x + 7)} + \frac{3(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})}{8ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{3*\frac{-5}{2}(2x + 1 + 0)}{8(x^{2} + x + 7)^{3}ln^{\frac{7}{2}}(x^{2} + x + 7)(x^{2} + x + 7)}\\=&\frac{-48x^{4}}{(x^{2} + x + 7)^{4}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{96x^{3}}{(x^{2} + x + 7)^{4}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{48x^{2}}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{44x^{4}}{(x^{2} + x + 7)^{4}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{88x^{3}}{(x^{2} + x + 7)^{4}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{72x^{2}}{(x^{2} + x + 7)^{4}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{48x}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{66x^{2}}{(x^{2} + x + 7)^{4}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{6}{(x^{2} + x + 7)^{2}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{36x^{2}}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{36x}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{36x^{4}}{(x^{2} + x + 7)^{4}ln^{\frac{5}{2}}(x^{2} + x + 7)} - \frac{72x^{3}}{(x^{2} + x + 7)^{4}ln^{\frac{5}{2}}(x^{2} + x + 7)} - \frac{54x^{2}}{(x^{2} + x + 7)^{4}ln^{\frac{5}{2}}(x^{2} + x + 7)} - \frac{24x}{(x^{2} + x + 7)^{4}ln^{\frac{1}{2}}(x^{2} + x + 7)} + \frac{12}{(x^{2} + x + 7)^{3}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{22x}{(x^{2} + x + 7)^{4}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{9}{(x^{2} + x + 7)^{3}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{18x}{(x^{2} + x + 7)^{4}ln^{\frac{5}{2}}(x^{2} + x + 7)} - \frac{3}{(x^{2} + x + 7)^{2}ln^{\frac{3}{2}}(x^{2} + x + 7)} + \frac{18x^{2}}{(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{18x}{(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} + \frac{9}{2(x^{2} + x + 7)^{3}ln^{\frac{5}{2}}(x^{2} + x + 7)} - \frac{15x^{4}}{(x^{2} + x + 7)^{4}ln^{\frac{7}{2}}(x^{2} + x + 7)} - \frac{30x^{3}}{(x^{2} + x + 7)^{4}ln^{\frac{7}{2}}(x^{2} + x + 7)} - \frac{45x^{2}}{2(x^{2} + x + 7)^{4}ln^{\frac{7}{2}}(x^{2} + x + 7)} - \frac{15x}{2(x^{2} + x + 7)^{4}ln^{\frac{7}{2}}(x^{2} + x + 7)} - \frac{3}{(x^{2} + x + 7)^{4}ln^{\frac{1}{2}}(x^{2} + x + 7)} - \frac{11}{4(x^{2} + x + 7)^{4}ln^{\frac{3}{2}}(x^{2} + x + 7)} - \frac{9}{4(x^{2} + x + 7)^{4}ln^{\frac{5}{2}}(x^{2} + x + 7)} - \frac{15}{16(x^{2} + x + 7)^{4}ln^{\frac{7}{2}}(x^{2} + x + 7)}\\ \end{split}\end{equation} \]