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。
\[ \begin{equation}\begin{split}【1/1】求函数(ln(x) - ln(sin(x))){\frac{1}{x}}^{2} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{ln(x)}{x^{2}} - \frac{ln(sin(x))}{x^{2}}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{ln(x)}{x^{2}} - \frac{ln(sin(x))}{x^{2}}\right)}{dx}\\=&\frac{-2ln(x)}{x^{3}} + \frac{1}{x^{2}(x)} - \frac{-2ln(sin(x))}{x^{3}} - \frac{cos(x)}{x^{2}(sin(x))}\\=&\frac{-2ln(x)}{x^{3}} + \frac{2ln(sin(x))}{x^{3}} - \frac{cos(x)}{x^{2}sin(x)} + \frac{1}{x^{3}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-2ln(x)}{x^{3}} + \frac{2ln(sin(x))}{x^{3}} - \frac{cos(x)}{x^{2}sin(x)} + \frac{1}{x^{3}}\right)}{dx}\\=&\frac{-2*-3ln(x)}{x^{4}} - \frac{2}{x^{3}(x)} + \frac{2*-3ln(sin(x))}{x^{4}} + \frac{2cos(x)}{x^{3}(sin(x))} - \frac{-2cos(x)}{x^{3}sin(x)} - \frac{-cos(x)cos(x)}{x^{2}sin^{2}(x)} - \frac{-sin(x)}{x^{2}sin(x)} + \frac{-3}{x^{4}}\\=&\frac{6ln(x)}{x^{4}} - \frac{6ln(sin(x))}{x^{4}} + \frac{4cos(x)}{x^{3}sin(x)} + \frac{cos^{2}(x)}{x^{2}sin^{2}(x)} - \frac{5}{x^{4}} + \frac{1}{x^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{6ln(x)}{x^{4}} - \frac{6ln(sin(x))}{x^{4}} + \frac{4cos(x)}{x^{3}sin(x)} + \frac{cos^{2}(x)}{x^{2}sin^{2}(x)} - \frac{5}{x^{4}} + \frac{1}{x^{2}}\right)}{dx}\\=&\frac{6*-4ln(x)}{x^{5}} + \frac{6}{x^{4}(x)} - \frac{6*-4ln(sin(x))}{x^{5}} - \frac{6cos(x)}{x^{4}(sin(x))} + \frac{4*-3cos(x)}{x^{4}sin(x)} + \frac{4*-cos(x)cos(x)}{x^{3}sin^{2}(x)} + \frac{4*-sin(x)}{x^{3}sin(x)} + \frac{-2cos^{2}(x)}{x^{3}sin^{2}(x)} + \frac{-2cos(x)cos^{2}(x)}{x^{2}sin^{3}(x)} + \frac{-2cos(x)sin(x)}{x^{2}sin^{2}(x)} - \frac{5*-4}{x^{5}} + \frac{-2}{x^{3}}\\=&\frac{-24ln(x)}{x^{5}} + \frac{24ln(sin(x))}{x^{5}} - \frac{18cos(x)}{x^{4}sin(x)} - \frac{6cos^{2}(x)}{x^{3}sin^{2}(x)} - \frac{2cos(x)}{x^{2}sin(x)} - \frac{2cos^{3}(x)}{x^{2}sin^{3}(x)} + \frac{26}{x^{5}} - \frac{6}{x^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-24ln(x)}{x^{5}} + \frac{24ln(sin(x))}{x^{5}} - \frac{18cos(x)}{x^{4}sin(x)} - \frac{6cos^{2}(x)}{x^{3}sin^{2}(x)} - \frac{2cos(x)}{x^{2}sin(x)} - \frac{2cos^{3}(x)}{x^{2}sin^{3}(x)} + \frac{26}{x^{5}} - \frac{6}{x^{3}}\right)}{dx}\\=&\frac{-24*-5ln(x)}{x^{6}} - \frac{24}{x^{5}(x)} + \frac{24*-5ln(sin(x))}{x^{6}} + \frac{24cos(x)}{x^{5}(sin(x))} - \frac{18*-4cos(x)}{x^{5}sin(x)} - \frac{18*-cos(x)cos(x)}{x^{4}sin^{2}(x)} - \frac{18*-sin(x)}{x^{4}sin(x)} - \frac{6*-3cos^{2}(x)}{x^{4}sin^{2}(x)} - \frac{6*-2cos(x)cos^{2}(x)}{x^{3}sin^{3}(x)} - \frac{6*-2cos(x)sin(x)}{x^{3}sin^{2}(x)} - \frac{2*-2cos(x)}{x^{3}sin(x)} - \frac{2*-cos(x)cos(x)}{x^{2}sin^{2}(x)} - \frac{2*-sin(x)}{x^{2}sin(x)} - \frac{2*-2cos^{3}(x)}{x^{3}sin^{3}(x)} - \frac{2*-3cos(x)cos^{3}(x)}{x^{2}sin^{4}(x)} - \frac{2*-3cos^{2}(x)sin(x)}{x^{2}sin^{3}(x)} + \frac{26*-5}{x^{6}} - \frac{6*-3}{x^{4}}\\=&\frac{120ln(x)}{x^{6}} - \frac{120ln(sin(x))}{x^{6}} + \frac{96cos(x)}{x^{5}sin(x)} + \frac{36cos^{2}(x)}{x^{4}sin^{2}(x)} + \frac{16cos(x)}{x^{3}sin(x)} + \frac{16cos^{3}(x)}{x^{3}sin^{3}(x)} + \frac{8cos^{2}(x)}{x^{2}sin^{2}(x)} + \frac{6cos^{4}(x)}{x^{2}sin^{4}(x)} + \frac{2}{x^{2}} - \frac{154}{x^{6}} + \frac{36}{x^{4}}\\ \end{split}\end{equation} \]

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