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。
\[ \begin{equation}\begin{split}【1/1】求函数(x - ln(x) - 1)(e^{x} - ln(1 + x))e^{x} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = - xe^{x}ln(x + 1) + xe^{{x}*{2}} + e^{x}ln(x + 1)ln(x) - e^{{x}*{2}}ln(x) + e^{x}ln(x + 1) - e^{{x}*{2}}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( - xe^{x}ln(x + 1) + xe^{{x}*{2}} + e^{x}ln(x + 1)ln(x) - e^{{x}*{2}}ln(x) + e^{x}ln(x + 1) - e^{{x}*{2}}\right)}{dx}\\=& - e^{x}ln(x + 1) - xe^{x}ln(x + 1) - \frac{xe^{x}(1 + 0)}{(x + 1)} + e^{{x}*{2}} + x*2e^{x}e^{x} + e^{x}ln(x + 1)ln(x) + \frac{e^{x}(1 + 0)ln(x)}{(x + 1)} + \frac{e^{x}ln(x + 1)}{(x)} - 2e^{x}e^{x}ln(x) - \frac{e^{{x}*{2}}}{(x)} + e^{x}ln(x + 1) + \frac{e^{x}(1 + 0)}{(x + 1)} - 2e^{x}e^{x}\\=& - xe^{x}ln(x + 1) - \frac{xe^{x}}{(x + 1)} + e^{x}ln(x + 1)ln(x) + \frac{e^{x}ln(x + 1)}{x} - 2e^{{x}*{2}}ln(x) + \frac{e^{x}ln(x)}{(x + 1)} + 2xe^{{x}*{2}} - e^{{x}*{2}} - \frac{e^{{x}*{2}}}{x} + \frac{e^{x}}{(x + 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( - xe^{x}ln(x + 1) - \frac{xe^{x}}{(x + 1)} + e^{x}ln(x + 1)ln(x) + \frac{e^{x}ln(x + 1)}{x} - 2e^{{x}*{2}}ln(x) + \frac{e^{x}ln(x)}{(x + 1)} + 2xe^{{x}*{2}} - e^{{x}*{2}} - \frac{e^{{x}*{2}}}{x} + \frac{e^{x}}{(x + 1)}\right)}{dx}\\=& - e^{x}ln(x + 1) - xe^{x}ln(x + 1) - \frac{xe^{x}(1 + 0)}{(x + 1)} - (\frac{-(1 + 0)}{(x + 1)^{2}})xe^{x} - \frac{e^{x}}{(x + 1)} - \frac{xe^{x}}{(x + 1)} + e^{x}ln(x + 1)ln(x) + \frac{e^{x}(1 + 0)ln(x)}{(x + 1)} + \frac{e^{x}ln(x + 1)}{(x)} + \frac{-e^{x}ln(x + 1)}{x^{2}} + \frac{e^{x}ln(x + 1)}{x} + \frac{e^{x}(1 + 0)}{x(x + 1)} - 2*2e^{x}e^{x}ln(x) - \frac{2e^{{x}*{2}}}{(x)} + (\frac{-(1 + 0)}{(x + 1)^{2}})e^{x}ln(x) + \frac{e^{x}ln(x)}{(x + 1)} + \frac{e^{x}}{(x + 1)(x)} + 2e^{{x}*{2}} + 2x*2e^{x}e^{x} - 2e^{x}e^{x} - \frac{-e^{{x}*{2}}}{x^{2}} - \frac{2e^{x}e^{x}}{x} + (\frac{-(1 + 0)}{(x + 1)^{2}})e^{x} + \frac{e^{x}}{(x + 1)}\\=&e^{x}ln(x + 1)ln(x) - xe^{x}ln(x + 1) - \frac{2xe^{x}}{(x + 1)} + \frac{xe^{x}}{(x + 1)^{2}} - e^{x}ln(x + 1) + \frac{2e^{x}ln(x)}{(x + 1)} + \frac{2e^{x}ln(x + 1)}{x} - \frac{e^{x}ln(x + 1)}{x^{2}} + \frac{2e^{x}}{(x + 1)x} - 4e^{{x}*{2}}ln(x) + 4xe^{{x}*{2}} - \frac{e^{x}ln(x)}{(x + 1)^{2}} - \frac{4e^{{x}*{2}}}{x} + \frac{e^{{x}*{2}}}{x^{2}} - \frac{e^{x}}{(x + 1)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( e^{x}ln(x + 1)ln(x) - xe^{x}ln(x + 1) - \frac{2xe^{x}}{(x + 1)} + \frac{xe^{x}}{(x + 1)^{2}} - e^{x}ln(x + 1) + \frac{2e^{x}ln(x)}{(x + 1)} + \frac{2e^{x}ln(x + 1)}{x} - \frac{e^{x}ln(x + 1)}{x^{2}} + \frac{2e^{x}}{(x + 1)x} - 4e^{{x}*{2}}ln(x) + 4xe^{{x}*{2}} - \frac{e^{x}ln(x)}{(x + 1)^{2}} - \frac{4e^{{x}*{2}}}{x} + \frac{e^{{x}*{2}}}{x^{2}} - \frac{e^{x}}{(x + 1)^{2}}\right)}{dx}\\=&e^{x}ln(x + 1)ln(x) + \frac{e^{x}(1 + 0)ln(x)}{(x + 1)} + \frac{e^{x}ln(x + 1)}{(x)} - e^{x}ln(x + 1) - xe^{x}ln(x + 1) - \frac{xe^{x}(1 + 0)}{(x + 1)} - 2(\frac{-(1 + 0)}{(x + 1)^{2}})xe^{x} - \frac{2e^{x}}{(x + 1)} - \frac{2xe^{x}}{(x + 1)} + (\frac{-2(1 + 0)}{(x + 1)^{3}})xe^{x} + \frac{e^{x}}{(x + 1)^{2}} + \frac{xe^{x}}{(x + 1)^{2}} - e^{x}ln(x + 1) - \frac{e^{x}(1 + 0)}{(x + 1)} + 2(\frac{-(1 + 0)}{(x + 1)^{2}})e^{x}ln(x) + \frac{2e^{x}ln(x)}{(x + 1)} + \frac{2e^{x}}{(x + 1)(x)} + \frac{2*-e^{x}ln(x + 1)}{x^{2}} + \frac{2e^{x}ln(x + 1)}{x} + \frac{2e^{x}(1 + 0)}{x(x + 1)} - \frac{-2e^{x}ln(x + 1)}{x^{3}} - \frac{e^{x}ln(x + 1)}{x^{2}} - \frac{e^{x}(1 + 0)}{x^{2}(x + 1)} + \frac{2(\frac{-(1 + 0)}{(x + 1)^{2}})e^{x}}{x} + \frac{2*-e^{x}}{(x + 1)x^{2}} + \frac{2e^{x}}{(x + 1)x} - 4*2e^{x}e^{x}ln(x) - \frac{4e^{{x}*{2}}}{(x)} + 4e^{{x}*{2}} + 4x*2e^{x}e^{x} - (\frac{-2(1 + 0)}{(x + 1)^{3}})e^{x}ln(x) - \frac{e^{x}ln(x)}{(x + 1)^{2}} - \frac{e^{x}}{(x + 1)^{2}(x)} - \frac{4*-e^{{x}*{2}}}{x^{2}} - \frac{4*2e^{x}e^{x}}{x} + \frac{-2e^{{x}*{2}}}{x^{3}} + \frac{2e^{x}e^{x}}{x^{2}} - (\frac{-2(1 + 0)}{(x + 1)^{3}})e^{x} - \frac{e^{x}}{(x + 1)^{2}}\\=&e^{x}ln(x + 1)ln(x) + \frac{3e^{x}ln(x)}{(x + 1)} + \frac{3e^{x}ln(x + 1)}{x} - 2e^{x}ln(x + 1) - xe^{x}ln(x + 1) - \frac{3xe^{x}}{(x + 1)} + \frac{3xe^{x}}{(x + 1)^{2}} - \frac{3e^{x}ln(x)}{(x + 1)^{2}} - \frac{2xe^{x}}{(x + 1)^{3}} + \frac{2e^{x}ln(x)}{(x + 1)^{3}} + \frac{6e^{x}}{(x + 1)x} - \frac{3e^{x}ln(x + 1)}{x^{2}} + \frac{2e^{x}ln(x + 1)}{x^{3}} - \frac{3e^{x}}{(x + 1)x^{2}} - \frac{3e^{x}}{(x + 1)^{2}x} - 8e^{{x}*{2}}ln(x) + 8xe^{{x}*{2}} + 4e^{{x}*{2}} - \frac{12e^{{x}*{2}}}{x} - \frac{3e^{x}}{(x + 1)} + \frac{6e^{{x}*{2}}}{x^{2}} - \frac{2e^{{x}*{2}}}{x^{3}} + \frac{2e^{x}}{(x + 1)^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( e^{x}ln(x + 1)ln(x) + \frac{3e^{x}ln(x)}{(x + 1)} + \frac{3e^{x}ln(x + 1)}{x} - 2e^{x}ln(x + 1) - xe^{x}ln(x + 1) - \frac{3xe^{x}}{(x + 1)} + \frac{3xe^{x}}{(x + 1)^{2}} - \frac{3e^{x}ln(x)}{(x + 1)^{2}} - \frac{2xe^{x}}{(x + 1)^{3}} + \frac{2e^{x}ln(x)}{(x + 1)^{3}} + \frac{6e^{x}}{(x + 1)x} - \frac{3e^{x}ln(x + 1)}{x^{2}} + \frac{2e^{x}ln(x + 1)}{x^{3}} - \frac{3e^{x}}{(x + 1)x^{2}} - \frac{3e^{x}}{(x + 1)^{2}x} - 8e^{{x}*{2}}ln(x) + 8xe^{{x}*{2}} + 4e^{{x}*{2}} - \frac{12e^{{x}*{2}}}{x} - \frac{3e^{x}}{(x + 1)} + \frac{6e^{{x}*{2}}}{x^{2}} - \frac{2e^{{x}*{2}}}{x^{3}} + \frac{2e^{x}}{(x + 1)^{3}}\right)}{dx}\\=&e^{x}ln(x + 1)ln(x) + \frac{e^{x}(1 + 0)ln(x)}{(x + 1)} + \frac{e^{x}ln(x + 1)}{(x)} + 3(\frac{-(1 + 0)}{(x + 1)^{2}})e^{x}ln(x) + \frac{3e^{x}ln(x)}{(x + 1)} + \frac{3e^{x}}{(x + 1)(x)} + \frac{3*-e^{x}ln(x + 1)}{x^{2}} + \frac{3e^{x}ln(x + 1)}{x} + \frac{3e^{x}(1 + 0)}{x(x + 1)} - 2e^{x}ln(x + 1) - \frac{2e^{x}(1 + 0)}{(x + 1)} - e^{x}ln(x + 1) - xe^{x}ln(x + 1) - \frac{xe^{x}(1 + 0)}{(x + 1)} - 3(\frac{-(1 + 0)}{(x + 1)^{2}})xe^{x} - \frac{3e^{x}}{(x + 1)} - \frac{3xe^{x}}{(x + 1)} + 3(\frac{-2(1 + 0)}{(x + 1)^{3}})xe^{x} + \frac{3e^{x}}{(x + 1)^{2}} + \frac{3xe^{x}}{(x + 1)^{2}} - 3(\frac{-2(1 + 0)}{(x + 1)^{3}})e^{x}ln(x) - \frac{3e^{x}ln(x)}{(x + 1)^{2}} - \frac{3e^{x}}{(x + 1)^{2}(x)} - 2(\frac{-3(1 + 0)}{(x + 1)^{4}})xe^{x} - \frac{2e^{x}}{(x + 1)^{3}} - \frac{2xe^{x}}{(x + 1)^{3}} + 2(\frac{-3(1 + 0)}{(x + 1)^{4}})e^{x}ln(x) + \frac{2e^{x}ln(x)}{(x + 1)^{3}} + \frac{2e^{x}}{(x + 1)^{3}(x)} + \frac{6(\frac{-(1 + 0)}{(x + 1)^{2}})e^{x}}{x} + \frac{6*-e^{x}}{(x + 1)x^{2}} + \frac{6e^{x}}{(x + 1)x} - \frac{3*-2e^{x}ln(x + 1)}{x^{3}} - \frac{3e^{x}ln(x + 1)}{x^{2}} - \frac{3e^{x}(1 + 0)}{x^{2}(x + 1)} + \frac{2*-3e^{x}ln(x + 1)}{x^{4}} + \frac{2e^{x}ln(x + 1)}{x^{3}} + \frac{2e^{x}(1 + 0)}{x^{3}(x + 1)} - \frac{3(\frac{-(1 + 0)}{(x + 1)^{2}})e^{x}}{x^{2}} - \frac{3*-2e^{x}}{(x + 1)x^{3}} - \frac{3e^{x}}{(x + 1)x^{2}} - \frac{3(\frac{-2(1 + 0)}{(x + 1)^{3}})e^{x}}{x} - \frac{3*-e^{x}}{(x + 1)^{2}x^{2}} - \frac{3e^{x}}{(x + 1)^{2}x} - 8*2e^{x}e^{x}ln(x) - \frac{8e^{{x}*{2}}}{(x)} + 8e^{{x}*{2}} + 8x*2e^{x}e^{x} + 4*2e^{x}e^{x} - \frac{12*-e^{{x}*{2}}}{x^{2}} - \frac{12*2e^{x}e^{x}}{x} - 3(\frac{-(1 + 0)}{(x + 1)^{2}})e^{x} - \frac{3e^{x}}{(x + 1)} + \frac{6*-2e^{{x}*{2}}}{x^{3}} + \frac{6*2e^{x}e^{x}}{x^{2}} - \frac{2*-3e^{{x}*{2}}}{x^{4}} - \frac{2*2e^{x}e^{x}}{x^{3}} + 2(\frac{-3(1 + 0)}{(x + 1)^{4}})e^{x} + \frac{2e^{x}}{(x + 1)^{3}}\\=&e^{x}ln(x + 1)ln(x) + \frac{4e^{x}ln(x)}{(x + 1)} + \frac{4e^{x}ln(x + 1)}{x} - \frac{6e^{x}ln(x)}{(x + 1)^{2}} + \frac{12e^{x}}{(x + 1)x} - \frac{6e^{x}ln(x + 1)}{x^{2}} - 3e^{x}ln(x + 1) - xe^{x}ln(x + 1) - \frac{4xe^{x}}{(x + 1)} + \frac{6xe^{x}}{(x + 1)^{2}} + \frac{8e^{x}ln(x)}{(x + 1)^{3}} - \frac{8xe^{x}}{(x + 1)^{3}} - \frac{6e^{x}ln(x)}{(x + 1)^{4}} - \frac{8e^{x}}{(x + 1)} - \frac{12e^{x}}{(x + 1)^{2}x} + \frac{6xe^{x}}{(x + 1)^{4}} + \frac{6e^{x}}{(x + 1)^{2}} + \frac{8e^{x}}{(x + 1)^{3}x} - \frac{12e^{x}}{(x + 1)x^{2}} + \frac{8e^{x}ln(x + 1)}{x^{3}} - \frac{6e^{x}ln(x + 1)}{x^{4}} + \frac{8e^{x}}{(x + 1)x^{3}} + \frac{6e^{x}}{(x + 1)^{2}x^{2}} - 16e^{{x}*{2}}ln(x) + 16xe^{{x}*{2}} + 16e^{{x}*{2}} - \frac{32e^{{x}*{2}}}{x} + \frac{24e^{{x}*{2}}}{x^{2}} - \frac{16e^{{x}*{2}}}{x^{3}} + \frac{6e^{{x}*{2}}}{x^{4}} - \frac{6e^{x}}{(x + 1)^{4}}\\ \end{split}\end{equation} \]

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