There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{x}{sqrt({x}^{(2 + {x}^{2})})}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{x}{sqrt({x}^{(x^{2} + 2)})}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x}{sqrt({x}^{(x^{2} + 2)})}\right)}{dx}\\=&\frac{1}{sqrt({x}^{(x^{2} + 2)})} + \frac{x*-({x}^{(x^{2} + 2)}((2x + 0)ln(x) + \frac{(x^{2} + 2)(1)}{(x)}))*\frac{1}{2}}{({x}^{(x^{2} + 2)})({x}^{(x^{2} + 2)})^{\frac{1}{2}}}\\=&\frac{1}{sqrt({x}^{(x^{2} + 2)})} - x^{2}{x}^{(2x^{2} + 4)}ln(x) - \frac{x^{2}{x}^{(2x^{2} + 4)}}{2} - {x}^{(2x^{2} + 4)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{sqrt({x}^{(x^{2} + 2)})} - x^{2}{x}^{(2x^{2} + 4)}ln(x) - \frac{x^{2}{x}^{(2x^{2} + 4)}}{2} - {x}^{(2x^{2} + 4)}\right)}{dx}\\=&\frac{-({x}^{(x^{2} + 2)}((2x + 0)ln(x) + \frac{(x^{2} + 2)(1)}{(x)}))*\frac{1}{2}}{({x}^{(x^{2} + 2)})({x}^{(x^{2} + 2)})^{\frac{1}{2}}} - 2x{x}^{(2x^{2} + 4)}ln(x) - x^{2}({x}^{(2x^{2} + 4)}((2*2x + 0)ln(x) + \frac{(2x^{2} + 4)(1)}{(x)}))ln(x) - \frac{x^{2}{x}^{(2x^{2} + 4)}}{(x)} - \frac{2x{x}^{(2x^{2} + 4)}}{2} - \frac{x^{2}({x}^{(2x^{2} + 4)}((2*2x + 0)ln(x) + \frac{(2x^{2} + 4)(1)}{(x)}))}{2} - ({x}^{(2x^{2} + 4)}((2*2x + 0)ln(x) + \frac{(2x^{2} + 4)(1)}{(x)}))\\=&-11x{x}^{(2x^{2} + 4)}ln(x) - 4x^{3}{x}^{(2x^{2} + 4)}ln^{2}(x) - 4x^{3}{x}^{(2x^{2} + 4)}ln(x) - \frac{5{x}^{(2x^{2} + 4)}}{x} - \frac{13x{x}^{(2x^{2} + 4)}}{2} - x^{3}{x}^{(2x^{2} + 4)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( -11x{x}^{(2x^{2} + 4)}ln(x) - 4x^{3}{x}^{(2x^{2} + 4)}ln^{2}(x) - 4x^{3}{x}^{(2x^{2} + 4)}ln(x) - \frac{5{x}^{(2x^{2} + 4)}}{x} - \frac{13x{x}^{(2x^{2} + 4)}}{2} - x^{3}{x}^{(2x^{2} + 4)}\right)}{dx}\\=&-11{x}^{(2x^{2} + 4)}ln(x) - 11x({x}^{(2x^{2} + 4)}((2*2x + 0)ln(x) + \frac{(2x^{2} + 4)(1)}{(x)}))ln(x) - \frac{11x{x}^{(2x^{2} + 4)}}{(x)} - 4*3x^{2}{x}^{(2x^{2} + 4)}ln^{2}(x) - 4x^{3}({x}^{(2x^{2} + 4)}((2*2x + 0)ln(x) + \frac{(2x^{2} + 4)(1)}{(x)}))ln^{2}(x) - \frac{4x^{3}{x}^{(2x^{2} + 4)}*2ln(x)}{(x)} - 4*3x^{2}{x}^{(2x^{2} + 4)}ln(x) - 4x^{3}({x}^{(2x^{2} + 4)}((2*2x + 0)ln(x) + \frac{(2x^{2} + 4)(1)}{(x)}))ln(x) - \frac{4x^{3}{x}^{(2x^{2} + 4)}}{(x)} - \frac{5*-{x}^{(2x^{2} + 4)}}{x^{2}} - \frac{5({x}^{(2x^{2} + 4)}((2*2x + 0)ln(x) + \frac{(2x^{2} + 4)(1)}{(x)}))}{x} - \frac{13{x}^{(2x^{2} + 4)}}{2} - \frac{13x({x}^{(2x^{2} + 4)}((2*2x + 0)ln(x) + \frac{(2x^{2} + 4)(1)}{(x)}))}{2} - 3x^{2}{x}^{(2x^{2} + 4)} - x^{3}({x}^{(2x^{2} + 4)}((2*2x + 0)ln(x) + \frac{(2x^{2} + 4)(1)}{(x)}))\\=&-75{x}^{(2x^{2} + 4)}ln(x) - 72x^{2}{x}^{(2x^{2} + 4)}ln^{2}(x) - 84x^{2}{x}^{(2x^{2} + 4)}ln(x) - \frac{107{x}^{(2x^{2} + 4)}}{2} - 16x^{4}{x}^{(2x^{2} + 4)}ln^{3}(x) - 24x^{4}{x}^{(2x^{2} + 4)}ln^{2}(x) - 12x^{4}{x}^{(2x^{2} + 4)}ln(x) - 24x^{2}{x}^{(2x^{2} + 4)} - \frac{15{x}^{(2x^{2} + 4)}}{x^{2}} - 2x^{4}{x}^{(2x^{2} + 4)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( -75{x}^{(2x^{2} + 4)}ln(x) - 72x^{2}{x}^{(2x^{2} + 4)}ln^{2}(x) - 84x^{2}{x}^{(2x^{2} + 4)}ln(x) - \frac{107{x}^{(2x^{2} + 4)}}{2} - 16x^{4}{x}^{(2x^{2} + 4)}ln^{3}(x) - 24x^{4}{x}^{(2x^{2} + 4)}ln^{2}(x) - 12x^{4}{x}^{(2x^{2} + 4)}ln(x) - 24x^{2}{x}^{(2x^{2} + 4)} - \frac{15{x}^{(2x^{2} + 4)}}{x^{2}} - 2x^{4}{x}^{(2x^{2} + 4)}\right)}{dx}\\=&-75({x}^{(2x^{2} + 4)}((2*2x + 0)ln(x) + \frac{(2x^{2} + 4)(1)}{(x)}))ln(x) - \frac{75{x}^{(2x^{2} + 4)}}{(x)} - 72*2x{x}^{(2x^{2} + 4)}ln^{2}(x) - 72x^{2}({x}^{(2x^{2} + 4)}((2*2x + 0)ln(x) + \frac{(2x^{2} + 4)(1)}{(x)}))ln^{2}(x) - \frac{72x^{2}{x}^{(2x^{2} + 4)}*2ln(x)}{(x)} - 84*2x{x}^{(2x^{2} + 4)}ln(x) - 84x^{2}({x}^{(2x^{2} + 4)}((2*2x + 0)ln(x) + \frac{(2x^{2} + 4)(1)}{(x)}))ln(x) - \frac{84x^{2}{x}^{(2x^{2} + 4)}}{(x)} - \frac{107({x}^{(2x^{2} + 4)}((2*2x + 0)ln(x) + \frac{(2x^{2} + 4)(1)}{(x)}))}{2} - 16*4x^{3}{x}^{(2x^{2} + 4)}ln^{3}(x) - 16x^{4}({x}^{(2x^{2} + 4)}((2*2x + 0)ln(x) + \frac{(2x^{2} + 4)(1)}{(x)}))ln^{3}(x) - \frac{16x^{4}{x}^{(2x^{2} + 4)}*3ln^{2}(x)}{(x)} - 24*4x^{3}{x}^{(2x^{2} + 4)}ln^{2}(x) - 24x^{4}({x}^{(2x^{2} + 4)}((2*2x + 0)ln(x) + \frac{(2x^{2} + 4)(1)}{(x)}))ln^{2}(x) - \frac{24x^{4}{x}^{(2x^{2} + 4)}*2ln(x)}{(x)} - 12*4x^{3}{x}^{(2x^{2} + 4)}ln(x) - 12x^{4}({x}^{(2x^{2} + 4)}((2*2x + 0)ln(x) + \frac{(2x^{2} + 4)(1)}{(x)}))ln(x) - \frac{12x^{4}{x}^{(2x^{2} + 4)}}{(x)} - 24*2x{x}^{(2x^{2} + 4)} - 24x^{2}({x}^{(2x^{2} + 4)}((2*2x + 0)ln(x) + \frac{(2x^{2} + 4)(1)}{(x)})) - \frac{15*-2{x}^{(2x^{2} + 4)}}{x^{3}} - \frac{15({x}^{(2x^{2} + 4)}((2*2x + 0)ln(x) + \frac{(2x^{2} + 4)(1)}{(x)}))}{x^{2}} - 2*4x^{3}{x}^{(2x^{2} + 4)} - 2x^{4}({x}^{(2x^{2} + 4)}((2*2x + 0)ln(x) + \frac{(2x^{2} + 4)(1)}{(x)}))\\=&-732x{x}^{(2x^{2} + 4)}ln^{2}(x) - 1012x{x}^{(2x^{2} + 4)}ln(x) - \frac{360{x}^{(2x^{2} + 4)}ln(x)}{x} - 416x^{3}{x}^{(2x^{2} + 4)}ln^{3}(x) - 720x^{3}{x}^{(2x^{2} + 4)}ln^{2}(x) - 408x^{3}{x}^{(2x^{2} + 4)}ln(x) - 64x^{5}{x}^{(2x^{2} + 4)}ln^{4}(x) - 128x^{5}{x}^{(2x^{2} + 4)}ln^{3}(x) - 96x^{5}{x}^{(2x^{2} + 4)}ln^{2}(x) - 32x^{5}{x}^{(2x^{2} + 4)}ln(x) - \frac{319{x}^{(2x^{2} + 4)}}{x} - 335x{x}^{(2x^{2} + 4)} - 76x^{3}{x}^{(2x^{2} + 4)} - \frac{30{x}^{(2x^{2} + 4)}}{x^{3}} - 4x^{5}{x}^{(2x^{2} + 4)}\\ \end{split}\end{equation} \]

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