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。
\[ \begin{equation}\begin{split}【1/1】求函数sin(x){\frac{1}{1}}^{cos(x)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = {1}^{cos(x)}sin(x)\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( {1}^{cos(x)}sin(x)\right)}{dx}\\=&({1}^{cos(x)}((-sin(x))ln(1) + \frac{(cos(x))(0)}{(1)}))sin(x) + {1}^{cos(x)}cos(x)\\=&-{1}^{cos(x)}ln(1)sin^{2}(x) + {1}^{cos(x)}cos(x)\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( -{1}^{cos(x)}ln(1)sin^{2}(x) + {1}^{cos(x)}cos(x)\right)}{dx}\\=&-({1}^{cos(x)}((-sin(x))ln(1) + \frac{(cos(x))(0)}{(1)}))ln(1)sin^{2}(x) - \frac{{1}^{cos(x)}*0sin^{2}(x)}{(1)} - {1}^{cos(x)}ln(1)*2sin(x)cos(x) + ({1}^{cos(x)}((-sin(x))ln(1) + \frac{(cos(x))(0)}{(1)}))cos(x) + {1}^{cos(x)}*-sin(x)\\=& - 3 * {1}^{cos(x)}ln(1)sin(x)cos(x) + {1}^{cos(x)}ln^{2}(1)sin^{3}(x) - {1}^{cos(x)}sin(x)\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( - 3 * {1}^{cos(x)}ln(1)sin(x)cos(x) + {1}^{cos(x)}ln^{2}(1)sin^{3}(x) - {1}^{cos(x)}sin(x)\right)}{dx}\\=& - 3({1}^{cos(x)}((-sin(x))ln(1) + \frac{(cos(x))(0)}{(1)}))ln(1)sin(x)cos(x) - \frac{3 * {1}^{cos(x)}*0sin(x)cos(x)}{(1)} - 3 * {1}^{cos(x)}ln(1)cos(x)cos(x) - 3 * {1}^{cos(x)}ln(1)sin(x)*-sin(x) + ({1}^{cos(x)}((-sin(x))ln(1) + \frac{(cos(x))(0)}{(1)}))ln^{2}(1)sin^{3}(x) + \frac{{1}^{cos(x)}*2ln(1)*0sin^{3}(x)}{(1)} + {1}^{cos(x)}ln^{2}(1)*3sin^{2}(x)cos(x) - ({1}^{cos(x)}((-sin(x))ln(1) + \frac{(cos(x))(0)}{(1)}))sin(x) - {1}^{cos(x)}cos(x)\\=&6 * {1}^{cos(x)}ln^{2}(1)sin^{2}(x)cos(x) - 3 * {1}^{cos(x)}ln(1)cos^{2}(x) - {1}^{cos(x)}ln^{3}(1)sin^{4}(x) + 4 * {1}^{cos(x)}ln(1)sin^{2}(x) - {1}^{cos(x)}cos(x)\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( 6 * {1}^{cos(x)}ln^{2}(1)sin^{2}(x)cos(x) - 3 * {1}^{cos(x)}ln(1)cos^{2}(x) - {1}^{cos(x)}ln^{3}(1)sin^{4}(x) + 4 * {1}^{cos(x)}ln(1)sin^{2}(x) - {1}^{cos(x)}cos(x)\right)}{dx}\\=&6({1}^{cos(x)}((-sin(x))ln(1) + \frac{(cos(x))(0)}{(1)}))ln^{2}(1)sin^{2}(x)cos(x) + \frac{6 * {1}^{cos(x)}*2ln(1)*0sin^{2}(x)cos(x)}{(1)} + 6 * {1}^{cos(x)}ln^{2}(1)*2sin(x)cos(x)cos(x) + 6 * {1}^{cos(x)}ln^{2}(1)sin^{2}(x)*-sin(x) - 3({1}^{cos(x)}((-sin(x))ln(1) + \frac{(cos(x))(0)}{(1)}))ln(1)cos^{2}(x) - \frac{3 * {1}^{cos(x)}*0cos^{2}(x)}{(1)} - 3 * {1}^{cos(x)}ln(1)*-2cos(x)sin(x) - ({1}^{cos(x)}((-sin(x))ln(1) + \frac{(cos(x))(0)}{(1)}))ln^{3}(1)sin^{4}(x) - \frac{{1}^{cos(x)}*3ln^{2}(1)*0sin^{4}(x)}{(1)} - {1}^{cos(x)}ln^{3}(1)*4sin^{3}(x)cos(x) + 4({1}^{cos(x)}((-sin(x))ln(1) + \frac{(cos(x))(0)}{(1)}))ln(1)sin^{2}(x) + \frac{4 * {1}^{cos(x)}*0sin^{2}(x)}{(1)} + 4 * {1}^{cos(x)}ln(1)*2sin(x)cos(x) - ({1}^{cos(x)}((-sin(x))ln(1) + \frac{(cos(x))(0)}{(1)}))cos(x) - {1}^{cos(x)}*-sin(x)\\=&-10 * {1}^{cos(x)}ln^{3}(1)sin^{3}(x)cos(x) + 15 * {1}^{cos(x)}ln^{2}(1)sin(x)cos^{2}(x) + 15 * {1}^{cos(x)}ln(1)sin(x)cos(x) + {1}^{cos(x)}ln^{4}(1)sin^{5}(x) - 10 * {1}^{cos(x)}ln^{2}(1)sin^{3}(x) + {1}^{cos(x)}sin(x)\\ \end{split}\end{equation} \]

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