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。
\[ \begin{equation}\begin{split}【1/1】求函数{x}^{cos(x)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( {x}^{cos(x)}\right)}{dx}\\=&({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))\\=&-{x}^{cos(x)}ln(x)sin(x) + \frac{{x}^{cos(x)}cos(x)}{x}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( -{x}^{cos(x)}ln(x)sin(x) + \frac{{x}^{cos(x)}cos(x)}{x}\right)}{dx}\\=&-({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))ln(x)sin(x) - \frac{{x}^{cos(x)}sin(x)}{(x)} - {x}^{cos(x)}ln(x)cos(x) + \frac{-{x}^{cos(x)}cos(x)}{x^{2}} + \frac{({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))cos(x)}{x} + \frac{{x}^{cos(x)}*-sin(x)}{x}\\=&{x}^{cos(x)}ln^{2}(x)sin^{2}(x) - \frac{2{x}^{cos(x)}ln(x)sin(x)cos(x)}{x} - \frac{2{x}^{cos(x)}sin(x)}{x} - {x}^{cos(x)}ln(x)cos(x) - \frac{{x}^{cos(x)}cos(x)}{x^{2}} + \frac{{x}^{cos(x)}cos^{2}(x)}{x^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( {x}^{cos(x)}ln^{2}(x)sin^{2}(x) - \frac{2{x}^{cos(x)}ln(x)sin(x)cos(x)}{x} - \frac{2{x}^{cos(x)}sin(x)}{x} - {x}^{cos(x)}ln(x)cos(x) - \frac{{x}^{cos(x)}cos(x)}{x^{2}} + \frac{{x}^{cos(x)}cos^{2}(x)}{x^{2}}\right)}{dx}\\=&({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))ln^{2}(x)sin^{2}(x) + \frac{{x}^{cos(x)}*2ln(x)sin^{2}(x)}{(x)} + {x}^{cos(x)}ln^{2}(x)*2sin(x)cos(x) - \frac{2*-{x}^{cos(x)}ln(x)sin(x)cos(x)}{x^{2}} - \frac{2({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))ln(x)sin(x)cos(x)}{x} - \frac{2{x}^{cos(x)}sin(x)cos(x)}{x(x)} - \frac{2{x}^{cos(x)}ln(x)cos(x)cos(x)}{x} - \frac{2{x}^{cos(x)}ln(x)sin(x)*-sin(x)}{x} - \frac{2*-{x}^{cos(x)}sin(x)}{x^{2}} - \frac{2({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))sin(x)}{x} - \frac{2{x}^{cos(x)}cos(x)}{x} - ({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))ln(x)cos(x) - \frac{{x}^{cos(x)}cos(x)}{(x)} - {x}^{cos(x)}ln(x)*-sin(x) - \frac{-2{x}^{cos(x)}cos(x)}{x^{3}} - \frac{({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))cos(x)}{x^{2}} - \frac{{x}^{cos(x)}*-sin(x)}{x^{2}} + \frac{-2{x}^{cos(x)}cos^{2}(x)}{x^{3}} + \frac{({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))cos^{2}(x)}{x^{2}} + \frac{{x}^{cos(x)}*-2cos(x)sin(x)}{x^{2}}\\=&3{x}^{cos(x)}ln^{2}(x)sin(x)cos(x) + \frac{3{x}^{cos(x)}ln^{2}(x)sin^{2}(x)cos(x)}{x} + \frac{3{x}^{cos(x)}ln(x)sin(x)cos(x)}{x^{2}} - {x}^{cos(x)}ln^{3}(x)sin^{3}(x) - \frac{3{x}^{cos(x)}ln(x)sin(x)cos^{2}(x)}{x^{2}} + \frac{6{x}^{cos(x)}ln(x)sin^{2}(x)}{x} - \frac{6{x}^{cos(x)}sin(x)cos(x)}{x^{2}} - \frac{3{x}^{cos(x)}ln(x)cos^{2}(x)}{x} + \frac{3{x}^{cos(x)}sin(x)}{x^{2}} - \frac{3{x}^{cos(x)}cos(x)}{x} + {x}^{cos(x)}ln(x)sin(x) + \frac{2{x}^{cos(x)}cos(x)}{x^{3}} - \frac{3{x}^{cos(x)}cos^{2}(x)}{x^{3}} + \frac{{x}^{cos(x)}cos^{3}(x)}{x^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( 3{x}^{cos(x)}ln^{2}(x)sin(x)cos(x) + \frac{3{x}^{cos(x)}ln^{2}(x)sin^{2}(x)cos(x)}{x} + \frac{3{x}^{cos(x)}ln(x)sin(x)cos(x)}{x^{2}} - {x}^{cos(x)}ln^{3}(x)sin^{3}(x) - \frac{3{x}^{cos(x)}ln(x)sin(x)cos^{2}(x)}{x^{2}} + \frac{6{x}^{cos(x)}ln(x)sin^{2}(x)}{x} - \frac{6{x}^{cos(x)}sin(x)cos(x)}{x^{2}} - \frac{3{x}^{cos(x)}ln(x)cos^{2}(x)}{x} + \frac{3{x}^{cos(x)}sin(x)}{x^{2}} - \frac{3{x}^{cos(x)}cos(x)}{x} + {x}^{cos(x)}ln(x)sin(x) + \frac{2{x}^{cos(x)}cos(x)}{x^{3}} - \frac{3{x}^{cos(x)}cos^{2}(x)}{x^{3}} + \frac{{x}^{cos(x)}cos^{3}(x)}{x^{3}}\right)}{dx}\\=&3({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))ln^{2}(x)sin(x)cos(x) + \frac{3{x}^{cos(x)}*2ln(x)sin(x)cos(x)}{(x)} + 3{x}^{cos(x)}ln^{2}(x)cos(x)cos(x) + 3{x}^{cos(x)}ln^{2}(x)sin(x)*-sin(x) + \frac{3*-{x}^{cos(x)}ln^{2}(x)sin^{2}(x)cos(x)}{x^{2}} + \frac{3({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))ln^{2}(x)sin^{2}(x)cos(x)}{x} + \frac{3{x}^{cos(x)}*2ln(x)sin^{2}(x)cos(x)}{x(x)} + \frac{3{x}^{cos(x)}ln^{2}(x)*2sin(x)cos(x)cos(x)}{x} + \frac{3{x}^{cos(x)}ln^{2}(x)sin^{2}(x)*-sin(x)}{x} + \frac{3*-2{x}^{cos(x)}ln(x)sin(x)cos(x)}{x^{3}} + \frac{3({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))ln(x)sin(x)cos(x)}{x^{2}} + \frac{3{x}^{cos(x)}sin(x)cos(x)}{x^{2}(x)} + \frac{3{x}^{cos(x)}ln(x)cos(x)cos(x)}{x^{2}} + \frac{3{x}^{cos(x)}ln(x)sin(x)*-sin(x)}{x^{2}} - ({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))ln^{3}(x)sin^{3}(x) - \frac{{x}^{cos(x)}*3ln^{2}(x)sin^{3}(x)}{(x)} - {x}^{cos(x)}ln^{3}(x)*3sin^{2}(x)cos(x) - \frac{3*-2{x}^{cos(x)}ln(x)sin(x)cos^{2}(x)}{x^{3}} - \frac{3({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))ln(x)sin(x)cos^{2}(x)}{x^{2}} - \frac{3{x}^{cos(x)}sin(x)cos^{2}(x)}{x^{2}(x)} - \frac{3{x}^{cos(x)}ln(x)cos(x)cos^{2}(x)}{x^{2}} - \frac{3{x}^{cos(x)}ln(x)sin(x)*-2cos(x)sin(x)}{x^{2}} + \frac{6*-{x}^{cos(x)}ln(x)sin^{2}(x)}{x^{2}} + \frac{6({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))ln(x)sin^{2}(x)}{x} + \frac{6{x}^{cos(x)}sin^{2}(x)}{x(x)} + \frac{6{x}^{cos(x)}ln(x)*2sin(x)cos(x)}{x} - \frac{6*-2{x}^{cos(x)}sin(x)cos(x)}{x^{3}} - \frac{6({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))sin(x)cos(x)}{x^{2}} - \frac{6{x}^{cos(x)}cos(x)cos(x)}{x^{2}} - \frac{6{x}^{cos(x)}sin(x)*-sin(x)}{x^{2}} - \frac{3*-{x}^{cos(x)}ln(x)cos^{2}(x)}{x^{2}} - \frac{3({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))ln(x)cos^{2}(x)}{x} - \frac{3{x}^{cos(x)}cos^{2}(x)}{x(x)} - \frac{3{x}^{cos(x)}ln(x)*-2cos(x)sin(x)}{x} + \frac{3*-2{x}^{cos(x)}sin(x)}{x^{3}} + \frac{3({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))sin(x)}{x^{2}} + \frac{3{x}^{cos(x)}cos(x)}{x^{2}} - \frac{3*-{x}^{cos(x)}cos(x)}{x^{2}} - \frac{3({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))cos(x)}{x} - \frac{3{x}^{cos(x)}*-sin(x)}{x} + ({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))ln(x)sin(x) + \frac{{x}^{cos(x)}sin(x)}{(x)} + {x}^{cos(x)}ln(x)cos(x) + \frac{2*-3{x}^{cos(x)}cos(x)}{x^{4}} + \frac{2({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))cos(x)}{x^{3}} + \frac{2{x}^{cos(x)}*-sin(x)}{x^{3}} - \frac{3*-3{x}^{cos(x)}cos^{2}(x)}{x^{4}} - \frac{3({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))cos^{2}(x)}{x^{3}} - \frac{3{x}^{cos(x)}*-2cos(x)sin(x)}{x^{3}} + \frac{-3{x}^{cos(x)}cos^{3}(x)}{x^{4}} + \frac{({x}^{cos(x)}((-sin(x))ln(x) + \frac{(cos(x))(1)}{(x)}))cos^{3}(x)}{x^{3}} + \frac{{x}^{cos(x)}*-3cos^{2}(x)sin(x)}{x^{3}}\\=&-6{x}^{cos(x)}ln^{3}(x)sin^{2}(x)cos(x) + \frac{12{x}^{cos(x)}ln^{2}(x)sin(x)cos^{2}(x)}{x} + \frac{28{x}^{cos(x)}ln(x)sin(x)cos(x)}{x} + 3{x}^{cos(x)}ln^{2}(x)cos^{2}(x) - \frac{6{x}^{cos(x)}ln^{2}(x)sin^{2}(x)cos(x)}{x^{2}} - \frac{4{x}^{cos(x)}ln^{3}(x)sin^{3}(x)cos(x)}{x} + \frac{6{x}^{cos(x)}ln^{2}(x)sin^{2}(x)cos^{2}(x)}{x^{2}} + \frac{24{x}^{cos(x)}ln(x)sin^{2}(x)cos(x)}{x^{2}} - \frac{8{x}^{cos(x)}ln(x)sin(x)cos(x)}{x^{3}} + \frac{12{x}^{cos(x)}ln(x)sin(x)cos^{2}(x)}{x^{3}} + \frac{24{x}^{cos(x)}sin(x)cos(x)}{x^{3}} + \frac{6{x}^{cos(x)}ln(x)cos^{2}(x)}{x^{2}} + {x}^{cos(x)}ln^{4}(x)sin^{4}(x) - \frac{4{x}^{cos(x)}ln(x)sin(x)cos^{3}(x)}{x^{3}} - 4{x}^{cos(x)}ln^{2}(x)sin^{2}(x) - \frac{12{x}^{cos(x)}ln(x)sin^{2}(x)}{x^{2}} - \frac{12{x}^{cos(x)}ln^{2}(x)sin^{3}(x)}{x} - \frac{12{x}^{cos(x)}sin(x)cos^{2}(x)}{x^{3}} - \frac{6{x}^{cos(x)}ln(x)cos^{3}(x)}{x^{2}} + \frac{12{x}^{cos(x)}sin^{2}(x)}{x^{2}} - \frac{12{x}^{cos(x)}cos^{2}(x)}{x^{2}} - \frac{8{x}^{cos(x)}sin(x)}{x^{3}} + \frac{6{x}^{cos(x)}cos(x)}{x^{2}} + {x}^{cos(x)}ln(x)cos(x) - \frac{6{x}^{cos(x)}cos(x)}{x^{4}} + \frac{11{x}^{cos(x)}cos^{2}(x)}{x^{4}} - \frac{6{x}^{cos(x)}cos^{3}(x)}{x^{4}} + \frac{4{x}^{cos(x)}sin(x)}{x} + \frac{{x}^{cos(x)}cos^{4}(x)}{x^{4}}\\ \end{split}\end{equation} \]

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