本次共计算 1 个题目：每一题对 x 求 4 阶导数。
    注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{{x}^{4}}{24} - \frac{11{(x + 1)}^{3}ln(x + 1)}{36} + \frac{{(x + 1)}^{3}{ln(x + 1)}^{2}}{12} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = - \frac{11}{36}x^{3}ln(x + 1) - \frac{11}{12}x^{2}ln(x + 1) - \frac{11}{12}xln(x + 1) + \frac{1}{12}x^{3}ln^{2}(x + 1) - \frac{11}{36}ln(x + 1) + \frac{1}{4}x^{2}ln^{2}(x + 1) + \frac{1}{4}xln^{2}(x + 1) + \frac{1}{24}x^{4} + \frac{1}{12}ln^{2}(x + 1)\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( - \frac{11}{36}x^{3}ln(x + 1) - \frac{11}{12}x^{2}ln(x + 1) - \frac{11}{12}xln(x + 1) + \frac{1}{12}x^{3}ln^{2}(x + 1) - \frac{11}{36}ln(x + 1) + \frac{1}{4}x^{2}ln^{2}(x + 1) + \frac{1}{4}xln^{2}(x + 1) + \frac{1}{24}x^{4} + \frac{1}{12}ln^{2}(x + 1)\right)}{dx}\\=& - \frac{11}{36}*3x^{2}ln(x + 1) - \frac{\frac{11}{36}x^{3}(1 + 0)}{(x + 1)} - \frac{11}{12}*2xln(x + 1) - \frac{\frac{11}{12}x^{2}(1 + 0)}{(x + 1)} - \frac{11}{12}ln(x + 1) - \frac{\frac{11}{12}x(1 + 0)}{(x + 1)} + \frac{1}{12}*3x^{2}ln^{2}(x + 1) + \frac{\frac{1}{12}x^{3}*2ln(x + 1)(1 + 0)}{(x + 1)} - \frac{\frac{11}{36}(1 + 0)}{(x + 1)} + \frac{1}{4}*2xln^{2}(x + 1) + \frac{\frac{1}{4}x^{2}*2ln(x + 1)(1 + 0)}{(x + 1)} + \frac{1}{4}ln^{2}(x + 1) + \frac{\frac{1}{4}x*2ln(x + 1)(1 + 0)}{(x + 1)} + \frac{1}{24}*4x^{3} + \frac{\frac{1}{12}*2ln(x + 1)(1 + 0)}{(x + 1)}\\=& - \frac{11x^{2}ln(x + 1)}{12} + \frac{x^{3}ln(x + 1)}{6(x + 1)} - \frac{11xln(x + 1)}{6} + \frac{x^{2}ln(x + 1)}{2(x + 1)} - \frac{11ln(x + 1)}{12} + \frac{xln(x + 1)}{2(x + 1)} + \frac{x^{2}ln^{2}(x + 1)}{4} - \frac{11x^{3}}{36(x + 1)} - \frac{11x^{2}}{12(x + 1)} + \frac{xln^{2}(x + 1)}{2} - \frac{11x}{12(x + 1)} + \frac{ln^{2}(x + 1)}{4} + \frac{ln(x + 1)}{6(x + 1)} + \frac{x^{3}}{6} - \frac{11}{36(x + 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( - \frac{11x^{2}ln(x + 1)}{12} + \frac{x^{3}ln(x + 1)}{6(x + 1)} - \frac{11xln(x + 1)}{6} + \frac{x^{2}ln(x + 1)}{2(x + 1)} - \frac{11ln(x + 1)}{12} + \frac{xln(x + 1)}{2(x + 1)} + \frac{x^{2}ln^{2}(x + 1)}{4} - \frac{11x^{3}}{36(x + 1)} - \frac{11x^{2}}{12(x + 1)} + \frac{xln^{2}(x + 1)}{2} - \frac{11x}{12(x + 1)} + \frac{ln^{2}(x + 1)}{4} + \frac{ln(x + 1)}{6(x + 1)} + \frac{x^{3}}{6} - \frac{11}{36(x + 1)}\right)}{dx}\\=& - \frac{11*2xln(x + 1)}{12} - \frac{11x^{2}(1 + 0)}{12(x + 1)} + \frac{(\frac{-(1 + 0)}{(x + 1)^{2}})x^{3}ln(x + 1)}{6} + \frac{3x^{2}ln(x + 1)}{6(x + 1)} + \frac{x^{3}(1 + 0)}{6(x + 1)(x + 1)} - \frac{11ln(x + 1)}{6} - \frac{11x(1 + 0)}{6(x + 1)} + \frac{(\frac{-(1 + 0)}{(x + 1)^{2}})x^{2}ln(x + 1)}{2} + \frac{2xln(x + 1)}{2(x + 1)} + \frac{x^{2}(1 + 0)}{2(x + 1)(x + 1)} - \frac{11(1 + 0)}{12(x + 1)} + \frac{(\frac{-(1 + 0)}{(x + 1)^{2}})xln(x + 1)}{2} + \frac{ln(x + 1)}{2(x + 1)} + \frac{x(1 + 0)}{2(x + 1)(x + 1)} + \frac{2xln^{2}(x + 1)}{4} + \frac{x^{2}*2ln(x + 1)(1 + 0)}{4(x + 1)} - \frac{11(\frac{-(1 + 0)}{(x + 1)^{2}})x^{3}}{36} - \frac{11*3x^{2}}{36(x + 1)} - \frac{11(\frac{-(1 + 0)}{(x + 1)^{2}})x^{2}}{12} - \frac{11*2x}{12(x + 1)} + \frac{ln^{2}(x + 1)}{2} + \frac{x*2ln(x + 1)(1 + 0)}{2(x + 1)} - \frac{11(\frac{-(1 + 0)}{(x + 1)^{2}})x}{12} - \frac{11}{12(x + 1)} + \frac{2ln(x + 1)(1 + 0)}{4(x + 1)} + \frac{(\frac{-(1 + 0)}{(x + 1)^{2}})ln(x + 1)}{6} + \frac{(1 + 0)}{6(x + 1)(x + 1)} + \frac{3x^{2}}{6} - \frac{11(\frac{-(1 + 0)}{(x + 1)^{2}})}{36}\\=& - \frac{11xln(x + 1)}{6} + \frac{x^{2}ln(x + 1)}{(x + 1)} - \frac{x^{3}ln(x + 1)}{6(x + 1)^{2}} + \frac{2xln(x + 1)}{(x + 1)} - \frac{x^{2}ln(x + 1)}{2(x + 1)^{2}} - \frac{11ln(x + 1)}{6} - \frac{xln(x + 1)}{2(x + 1)^{2}} - \frac{11x}{3(x + 1)} - \frac{11x^{2}}{6(x + 1)} + \frac{17x^{2}}{12(x + 1)^{2}} + \frac{17x}{12(x + 1)^{2}} + \frac{17x^{3}}{36(x + 1)^{2}} + \frac{ln(x + 1)}{(x + 1)} + \frac{xln^{2}(x + 1)}{2} - \frac{ln(x + 1)}{6(x + 1)^{2}} + \frac{ln^{2}(x + 1)}{2} + \frac{17}{36(x + 1)^{2}} - \frac{11}{6(x + 1)} + \frac{x^{2}}{2}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( - \frac{11xln(x + 1)}{6} + \frac{x^{2}ln(x + 1)}{(x + 1)} - \frac{x^{3}ln(x + 1)}{6(x + 1)^{2}} + \frac{2xln(x + 1)}{(x + 1)} - \frac{x^{2}ln(x + 1)}{2(x + 1)^{2}} - \frac{11ln(x + 1)}{6} - \frac{xln(x + 1)}{2(x + 1)^{2}} - \frac{11x}{3(x + 1)} - \frac{11x^{2}}{6(x + 1)} + \frac{17x^{2}}{12(x + 1)^{2}} + \frac{17x}{12(x + 1)^{2}} + \frac{17x^{3}}{36(x + 1)^{2}} + \frac{ln(x + 1)}{(x + 1)} + \frac{xln^{2}(x + 1)}{2} - \frac{ln(x + 1)}{6(x + 1)^{2}} + \frac{ln^{2}(x + 1)}{2} + \frac{17}{36(x + 1)^{2}} - \frac{11}{6(x + 1)} + \frac{x^{2}}{2}\right)}{dx}\\=& - \frac{11ln(x + 1)}{6} - \frac{11x(1 + 0)}{6(x + 1)} + (\frac{-(1 + 0)}{(x + 1)^{2}})x^{2}ln(x + 1) + \frac{2xln(x + 1)}{(x + 1)} + \frac{x^{2}(1 + 0)}{(x + 1)(x + 1)} - \frac{(\frac{-2(1 + 0)}{(x + 1)^{3}})x^{3}ln(x + 1)}{6} - \frac{3x^{2}ln(x + 1)}{6(x + 1)^{2}} - \frac{x^{3}(1 + 0)}{6(x + 1)^{2}(x + 1)} + 2(\frac{-(1 + 0)}{(x + 1)^{2}})xln(x + 1) + \frac{2ln(x + 1)}{(x + 1)} + \frac{2x(1 + 0)}{(x + 1)(x + 1)} - \frac{(\frac{-2(1 + 0)}{(x + 1)^{3}})x^{2}ln(x + 1)}{2} - \frac{2xln(x + 1)}{2(x + 1)^{2}} - \frac{x^{2}(1 + 0)}{2(x + 1)^{2}(x + 1)} - \frac{11(1 + 0)}{6(x + 1)} - \frac{(\frac{-2(1 + 0)}{(x + 1)^{3}})xln(x + 1)}{2} - \frac{ln(x + 1)}{2(x + 1)^{2}} - \frac{x(1 + 0)}{2(x + 1)^{2}(x + 1)} - \frac{11(\frac{-(1 + 0)}{(x + 1)^{2}})x}{3} - \frac{11}{3(x + 1)} - \frac{11(\frac{-(1 + 0)}{(x + 1)^{2}})x^{2}}{6} - \frac{11*2x}{6(x + 1)} + \frac{17(\frac{-2(1 + 0)}{(x + 1)^{3}})x^{2}}{12} + \frac{17*2x}{12(x + 1)^{2}} + \frac{17(\frac{-2(1 + 0)}{(x + 1)^{3}})x}{12} + \frac{17}{12(x + 1)^{2}} + \frac{17(\frac{-2(1 + 0)}{(x + 1)^{3}})x^{3}}{36} + \frac{17*3x^{2}}{36(x + 1)^{2}} + (\frac{-(1 + 0)}{(x + 1)^{2}})ln(x + 1) + \frac{(1 + 0)}{(x + 1)(x + 1)} + \frac{ln^{2}(x + 1)}{2} + \frac{x*2ln(x + 1)(1 + 0)}{2(x + 1)} - \frac{(\frac{-2(1 + 0)}{(x + 1)^{3}})ln(x + 1)}{6} - \frac{(1 + 0)}{6(x + 1)^{2}(x + 1)} + \frac{2ln(x + 1)(1 + 0)}{2(x + 1)} + \frac{17(\frac{-2(1 + 0)}{(x + 1)^{3}})}{36} - \frac{11(\frac{-(1 + 0)}{(x + 1)^{2}})}{6} + \frac{2x}{2}\\=& - \frac{11ln(x + 1)}{6} - \frac{3x^{2}ln(x + 1)}{2(x + 1)^{2}} + \frac{3xln(x + 1)}{(x + 1)} - \frac{3xln(x + 1)}{(x + 1)^{2}} + \frac{x^{3}ln(x + 1)}{3(x + 1)^{3}} + \frac{x^{2}ln(x + 1)}{(x + 1)^{3}} + \frac{xln(x + 1)}{(x + 1)^{3}} + \frac{3ln(x + 1)}{(x + 1)} + \frac{17x}{2(x + 1)^{2}} + \frac{17x^{2}}{4(x + 1)^{2}} - \frac{11x}{2(x + 1)} - \frac{10x^{2}}{3(x + 1)^{3}} - \frac{10x}{3(x + 1)^{3}} - \frac{10x^{3}}{9(x + 1)^{3}} - \frac{3ln(x + 1)}{2(x + 1)^{2}} + \frac{ln(x + 1)}{3(x + 1)^{3}} - \frac{10}{9(x + 1)^{3}} + \frac{17}{4(x + 1)^{2}} + \frac{ln^{2}(x + 1)}{2} - \frac{11}{2(x + 1)} + x\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( - \frac{11ln(x + 1)}{6} - \frac{3x^{2}ln(x + 1)}{2(x + 1)^{2}} + \frac{3xln(x + 1)}{(x + 1)} - \frac{3xln(x + 1)}{(x + 1)^{2}} + \frac{x^{3}ln(x + 1)}{3(x + 1)^{3}} + \frac{x^{2}ln(x + 1)}{(x + 1)^{3}} + \frac{xln(x + 1)}{(x + 1)^{3}} + \frac{3ln(x + 1)}{(x + 1)} + \frac{17x}{2(x + 1)^{2}} + \frac{17x^{2}}{4(x + 1)^{2}} - \frac{11x}{2(x + 1)} - \frac{10x^{2}}{3(x + 1)^{3}} - \frac{10x}{3(x + 1)^{3}} - \frac{10x^{3}}{9(x + 1)^{3}} - \frac{3ln(x + 1)}{2(x + 1)^{2}} + \frac{ln(x + 1)}{3(x + 1)^{3}} - \frac{10}{9(x + 1)^{3}} + \frac{17}{4(x + 1)^{2}} + \frac{ln^{2}(x + 1)}{2} - \frac{11}{2(x + 1)} + x\right)}{dx}\\=& - \frac{11(1 + 0)}{6(x + 1)} - \frac{3(\frac{-2(1 + 0)}{(x + 1)^{3}})x^{2}ln(x + 1)}{2} - \frac{3*2xln(x + 1)}{2(x + 1)^{2}} - \frac{3x^{2}(1 + 0)}{2(x + 1)^{2}(x + 1)} + 3(\frac{-(1 + 0)}{(x + 1)^{2}})xln(x + 1) + \frac{3ln(x + 1)}{(x + 1)} + \frac{3x(1 + 0)}{(x + 1)(x + 1)} - 3(\frac{-2(1 + 0)}{(x + 1)^{3}})xln(x + 1) - \frac{3ln(x + 1)}{(x + 1)^{2}} - \frac{3x(1 + 0)}{(x + 1)^{2}(x + 1)} + \frac{(\frac{-3(1 + 0)}{(x + 1)^{4}})x^{3}ln(x + 1)}{3} + \frac{3x^{2}ln(x + 1)}{3(x + 1)^{3}} + \frac{x^{3}(1 + 0)}{3(x + 1)^{3}(x + 1)} + (\frac{-3(1 + 0)}{(x + 1)^{4}})x^{2}ln(x + 1) + \frac{2xln(x + 1)}{(x + 1)^{3}} + \frac{x^{2}(1 + 0)}{(x + 1)^{3}(x + 1)} + (\frac{-3(1 + 0)}{(x + 1)^{4}})xln(x + 1) + \frac{ln(x + 1)}{(x + 1)^{3}} + \frac{x(1 + 0)}{(x + 1)^{3}(x + 1)} + 3(\frac{-(1 + 0)}{(x + 1)^{2}})ln(x + 1) + \frac{3(1 + 0)}{(x + 1)(x + 1)} + \frac{17(\frac{-2(1 + 0)}{(x + 1)^{3}})x}{2} + \frac{17}{2(x + 1)^{2}} + \frac{17(\frac{-2(1 + 0)}{(x + 1)^{3}})x^{2}}{4} + \frac{17*2x}{4(x + 1)^{2}} - \frac{11(\frac{-(1 + 0)}{(x + 1)^{2}})x}{2} - \frac{11}{2(x + 1)} - \frac{10(\frac{-3(1 + 0)}{(x + 1)^{4}})x^{2}}{3} - \frac{10*2x}{3(x + 1)^{3}} - \frac{10(\frac{-3(1 + 0)}{(x + 1)^{4}})x}{3} - \frac{10}{3(x + 1)^{3}} - \frac{10(\frac{-3(1 + 0)}{(x + 1)^{4}})x^{3}}{9} - \frac{10*3x^{2}}{9(x + 1)^{3}} - \frac{3(\frac{-2(1 + 0)}{(x + 1)^{3}})ln(x + 1)}{2} - \frac{3(1 + 0)}{2(x + 1)^{2}(x + 1)} + \frac{(\frac{-3(1 + 0)}{(x + 1)^{4}})ln(x + 1)}{3} + \frac{(1 + 0)}{3(x + 1)^{3}(x + 1)} - \frac{10(\frac{-3(1 + 0)}{(x + 1)^{4}})}{9} + \frac{17(\frac{-2(1 + 0)}{(x + 1)^{3}})}{4} + \frac{2ln(x + 1)(1 + 0)}{2(x + 1)} - \frac{11(\frac{-(1 + 0)}{(x + 1)^{2}})}{2} + 1\\=& - \frac{6xln(x + 1)}{(x + 1)^{2}} + \frac{4x^{2}ln(x + 1)}{(x + 1)^{3}} + \frac{8xln(x + 1)}{(x + 1)^{3}} - \frac{x^{3}ln(x + 1)}{(x + 1)^{4}} + \frac{4ln(x + 1)}{(x + 1)} - \frac{3x^{2}ln(x + 1)}{(x + 1)^{4}} - \frac{6ln(x + 1)}{(x + 1)^{2}} - \frac{3xln(x + 1)}{(x + 1)^{4}} - \frac{40x^{2}}{3(x + 1)^{3}} + \frac{17x}{(x + 1)^{2}} - \frac{80x}{3(x + 1)^{3}} + \frac{11x^{3}}{3(x + 1)^{4}} + \frac{11x^{2}}{(x + 1)^{4}} + \frac{4ln(x + 1)}{(x + 1)^{3}} + \frac{11x}{(x + 1)^{4}} - \frac{ln(x + 1)}{(x + 1)^{4}} + \frac{11}{3(x + 1)^{4}} - \frac{40}{3(x + 1)^{3}} + \frac{17}{(x + 1)^{2}} - \frac{22}{3(x + 1)} + 1\\ \end{split}\end{equation} \]

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