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。
\[ \begin{equation}\begin{split}【1/1】求函数{x}^{4}{{e}^{x}}^{ln(x)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = x^{4}{{e}^{x}}^{ln(x)}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( x^{4}{{e}^{x}}^{ln(x)}\right)}{dx}\\=&4x^{3}{{e}^{x}}^{ln(x)} + x^{4}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))\\=&x^{3}{{e}^{x}}^{ln(x)}ln({e}^{x}) + x^{4}{{e}^{x}}^{ln(x)}ln(x) + 4x^{3}{{e}^{x}}^{ln(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( x^{3}{{e}^{x}}^{ln(x)}ln({e}^{x}) + x^{4}{{e}^{x}}^{ln(x)}ln(x) + 4x^{3}{{e}^{x}}^{ln(x)}\right)}{dx}\\=&3x^{2}{{e}^{x}}^{ln(x)}ln({e}^{x}) + x^{3}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))ln({e}^{x}) + \frac{x^{3}{{e}^{x}}^{ln(x)}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{({e}^{x})} + 4x^{3}{{e}^{x}}^{ln(x)}ln(x) + x^{4}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))ln(x) + \frac{x^{4}{{e}^{x}}^{ln(x)}}{(x)} + 4*3x^{2}{{e}^{x}}^{ln(x)} + 4x^{3}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))\\=&x^{3}{{e}^{x}}^{ln(x)}ln({e}^{x})ln(x) + x^{3}{{e}^{x}}^{ln(x)}ln(x)ln({e}^{x}) + x^{2}{{e}^{x}}^{ln(x)}ln^{2}({e}^{x}) + x^{4}{{e}^{x}}^{ln(x)}ln^{2}(x) + 8x^{3}{{e}^{x}}^{ln(x)}ln(x) + 7x^{2}{{e}^{x}}^{ln(x)}ln({e}^{x}) + 2x^{3}{{e}^{x}}^{ln(x)} + 12x^{2}{{e}^{x}}^{ln(x)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( x^{3}{{e}^{x}}^{ln(x)}ln({e}^{x})ln(x) + x^{3}{{e}^{x}}^{ln(x)}ln(x)ln({e}^{x}) + x^{2}{{e}^{x}}^{ln(x)}ln^{2}({e}^{x}) + x^{4}{{e}^{x}}^{ln(x)}ln^{2}(x) + 8x^{3}{{e}^{x}}^{ln(x)}ln(x) + 7x^{2}{{e}^{x}}^{ln(x)}ln({e}^{x}) + 2x^{3}{{e}^{x}}^{ln(x)} + 12x^{2}{{e}^{x}}^{ln(x)}\right)}{dx}\\=&3x^{2}{{e}^{x}}^{ln(x)}ln({e}^{x})ln(x) + x^{3}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))ln({e}^{x})ln(x) + \frac{x^{3}{{e}^{x}}^{ln(x)}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(x)}{({e}^{x})} + \frac{x^{3}{{e}^{x}}^{ln(x)}ln({e}^{x})}{(x)} + 3x^{2}{{e}^{x}}^{ln(x)}ln(x)ln({e}^{x}) + x^{3}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))ln(x)ln({e}^{x}) + \frac{x^{3}{{e}^{x}}^{ln(x)}ln({e}^{x})}{(x)} + \frac{x^{3}{{e}^{x}}^{ln(x)}ln(x)({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{({e}^{x})} + 2x{{e}^{x}}^{ln(x)}ln^{2}({e}^{x}) + x^{2}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))ln^{2}({e}^{x}) + \frac{x^{2}{{e}^{x}}^{ln(x)}*2ln({e}^{x})({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{({e}^{x})} + 4x^{3}{{e}^{x}}^{ln(x)}ln^{2}(x) + x^{4}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))ln^{2}(x) + \frac{x^{4}{{e}^{x}}^{ln(x)}*2ln(x)}{(x)} + 8*3x^{2}{{e}^{x}}^{ln(x)}ln(x) + 8x^{3}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))ln(x) + \frac{8x^{3}{{e}^{x}}^{ln(x)}}{(x)} + 7*2x{{e}^{x}}^{ln(x)}ln({e}^{x}) + 7x^{2}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))ln({e}^{x}) + \frac{7x^{2}{{e}^{x}}^{ln(x)}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{({e}^{x})} + 2*3x^{2}{{e}^{x}}^{ln(x)} + 2x^{3}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})})) + 12*2x{{e}^{x}}^{ln(x)} + 12x^{2}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))\\=&11x^{2}{{e}^{x}}^{ln(x)}ln({e}^{x})ln(x) + 2x^{2}{{e}^{x}}^{ln(x)}ln^{2}({e}^{x})ln(x) + 2x^{3}{{e}^{x}}^{ln(x)}ln^{2}(x)ln({e}^{x}) + x^{2}{{e}^{x}}^{ln(x)}ln(x)ln^{2}({e}^{x}) + 10x^{2}{{e}^{x}}^{ln(x)}ln(x)ln({e}^{x}) + x^{3}{{e}^{x}}^{ln(x)}ln({e}^{x})ln^{2}(x) + 9x{{e}^{x}}^{ln(x)}ln^{2}({e}^{x}) + x{{e}^{x}}^{ln(x)}ln^{3}({e}^{x}) + 6x^{2}{{e}^{x}}^{ln(x)}ln({e}^{x}) + 12x^{3}{{e}^{x}}^{ln(x)}ln^{2}(x) + x^{4}{{e}^{x}}^{ln(x)}ln^{3}(x) + 6x^{3}{{e}^{x}}^{ln(x)}ln(x) + 36x^{2}{{e}^{x}}^{ln(x)}ln(x) + 26x{{e}^{x}}^{ln(x)}ln({e}^{x}) + 21x^{2}{{e}^{x}}^{ln(x)} + 24x{{e}^{x}}^{ln(x)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( 11x^{2}{{e}^{x}}^{ln(x)}ln({e}^{x})ln(x) + 2x^{2}{{e}^{x}}^{ln(x)}ln^{2}({e}^{x})ln(x) + 2x^{3}{{e}^{x}}^{ln(x)}ln^{2}(x)ln({e}^{x}) + x^{2}{{e}^{x}}^{ln(x)}ln(x)ln^{2}({e}^{x}) + 10x^{2}{{e}^{x}}^{ln(x)}ln(x)ln({e}^{x}) + x^{3}{{e}^{x}}^{ln(x)}ln({e}^{x})ln^{2}(x) + 9x{{e}^{x}}^{ln(x)}ln^{2}({e}^{x}) + x{{e}^{x}}^{ln(x)}ln^{3}({e}^{x}) + 6x^{2}{{e}^{x}}^{ln(x)}ln({e}^{x}) + 12x^{3}{{e}^{x}}^{ln(x)}ln^{2}(x) + x^{4}{{e}^{x}}^{ln(x)}ln^{3}(x) + 6x^{3}{{e}^{x}}^{ln(x)}ln(x) + 36x^{2}{{e}^{x}}^{ln(x)}ln(x) + 26x{{e}^{x}}^{ln(x)}ln({e}^{x}) + 21x^{2}{{e}^{x}}^{ln(x)} + 24x{{e}^{x}}^{ln(x)}\right)}{dx}\\=&11*2x{{e}^{x}}^{ln(x)}ln({e}^{x})ln(x) + 11x^{2}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))ln({e}^{x})ln(x) + \frac{11x^{2}{{e}^{x}}^{ln(x)}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(x)}{({e}^{x})} + \frac{11x^{2}{{e}^{x}}^{ln(x)}ln({e}^{x})}{(x)} + 2*2x{{e}^{x}}^{ln(x)}ln^{2}({e}^{x})ln(x) + 2x^{2}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))ln^{2}({e}^{x})ln(x) + \frac{2x^{2}{{e}^{x}}^{ln(x)}*2ln({e}^{x})({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln(x)}{({e}^{x})} + \frac{2x^{2}{{e}^{x}}^{ln(x)}ln^{2}({e}^{x})}{(x)} + 2*3x^{2}{{e}^{x}}^{ln(x)}ln^{2}(x)ln({e}^{x}) + 2x^{3}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))ln^{2}(x)ln({e}^{x}) + \frac{2x^{3}{{e}^{x}}^{ln(x)}*2ln(x)ln({e}^{x})}{(x)} + \frac{2x^{3}{{e}^{x}}^{ln(x)}ln^{2}(x)({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{({e}^{x})} + 2x{{e}^{x}}^{ln(x)}ln(x)ln^{2}({e}^{x}) + x^{2}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))ln(x)ln^{2}({e}^{x}) + \frac{x^{2}{{e}^{x}}^{ln(x)}ln^{2}({e}^{x})}{(x)} + \frac{x^{2}{{e}^{x}}^{ln(x)}ln(x)*2ln({e}^{x})({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{({e}^{x})} + 10*2x{{e}^{x}}^{ln(x)}ln(x)ln({e}^{x}) + 10x^{2}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))ln(x)ln({e}^{x}) + \frac{10x^{2}{{e}^{x}}^{ln(x)}ln({e}^{x})}{(x)} + \frac{10x^{2}{{e}^{x}}^{ln(x)}ln(x)({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{({e}^{x})} + 3x^{2}{{e}^{x}}^{ln(x)}ln({e}^{x})ln^{2}(x) + x^{3}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))ln({e}^{x})ln^{2}(x) + \frac{x^{3}{{e}^{x}}^{ln(x)}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))ln^{2}(x)}{({e}^{x})} + \frac{x^{3}{{e}^{x}}^{ln(x)}ln({e}^{x})*2ln(x)}{(x)} + 9{{e}^{x}}^{ln(x)}ln^{2}({e}^{x}) + 9x({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))ln^{2}({e}^{x}) + \frac{9x{{e}^{x}}^{ln(x)}*2ln({e}^{x})({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{({e}^{x})} + {{e}^{x}}^{ln(x)}ln^{3}({e}^{x}) + x({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))ln^{3}({e}^{x}) + \frac{x{{e}^{x}}^{ln(x)}*3ln^{2}({e}^{x})({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{({e}^{x})} + 6*2x{{e}^{x}}^{ln(x)}ln({e}^{x}) + 6x^{2}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))ln({e}^{x}) + \frac{6x^{2}{{e}^{x}}^{ln(x)}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{({e}^{x})} + 12*3x^{2}{{e}^{x}}^{ln(x)}ln^{2}(x) + 12x^{3}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))ln^{2}(x) + \frac{12x^{3}{{e}^{x}}^{ln(x)}*2ln(x)}{(x)} + 4x^{3}{{e}^{x}}^{ln(x)}ln^{3}(x) + x^{4}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))ln^{3}(x) + \frac{x^{4}{{e}^{x}}^{ln(x)}*3ln^{2}(x)}{(x)} + 6*3x^{2}{{e}^{x}}^{ln(x)}ln(x) + 6x^{3}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))ln(x) + \frac{6x^{3}{{e}^{x}}^{ln(x)}}{(x)} + 36*2x{{e}^{x}}^{ln(x)}ln(x) + 36x^{2}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))ln(x) + \frac{36x^{2}{{e}^{x}}^{ln(x)}}{(x)} + 26{{e}^{x}}^{ln(x)}ln({e}^{x}) + 26x({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))ln({e}^{x}) + \frac{26x{{e}^{x}}^{ln(x)}({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{({e}^{x})} + 21*2x{{e}^{x}}^{ln(x)} + 21x^{2}({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})})) + 24{{e}^{x}}^{ln(x)} + 24x({{e}^{x}}^{ln(x)}((\frac{1}{(x)})ln({e}^{x}) + \frac{(ln(x))(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})))}{({e}^{x})}))\\=&58x{{e}^{x}}^{ln(x)}ln({e}^{x})ln(x) + 25x{{e}^{x}}^{ln(x)}ln^{2}({e}^{x})ln(x) + 27x^{2}{{e}^{x}}^{ln(x)}ln^{2}(x)ln({e}^{x}) + 12x^{2}{{e}^{x}}^{ln(x)}ln({e}^{x})ln(x) + 3x{{e}^{x}}^{ln(x)}ln^{3}({e}^{x})ln(x) + 3x^{2}{{e}^{x}}^{ln(x)}ln^{2}(x)ln^{2}({e}^{x}) + 3x^{3}{{e}^{x}}^{ln(x)}ln^{3}(x)ln({e}^{x}) + 12x^{2}{{e}^{x}}^{ln(x)}ln(x)ln({e}^{x}) + 11x{{e}^{x}}^{ln(x)}ln(x)ln^{2}({e}^{x}) + 3x^{2}{{e}^{x}}^{ln(x)}ln^{2}({e}^{x})ln^{2}(x) + 46x{{e}^{x}}^{ln(x)}ln(x)ln({e}^{x}) + x{{e}^{x}}^{ln(x)}ln(x)ln^{3}({e}^{x}) + 15x^{2}{{e}^{x}}^{ln(x)}ln({e}^{x})ln^{2}(x) + x^{3}{{e}^{x}}^{ln(x)}ln({e}^{x})ln^{3}(x) + 35{{e}^{x}}^{ln(x)}ln^{2}({e}^{x}) + 10{{e}^{x}}^{ln(x)}ln^{3}({e}^{x}) + {{e}^{x}}^{ln(x)}ln^{4}({e}^{x}) + 12x^{3}{{e}^{x}}^{ln(x)}ln^{2}(x) + 12x{{e}^{x}}^{ln(x)}ln^{2}({e}^{x}) + 84x^{2}{{e}^{x}}^{ln(x)}ln(x) + 16x^{3}{{e}^{x}}^{ln(x)}ln^{3}(x) + 72x^{2}{{e}^{x}}^{ln(x)}ln^{2}(x) + x^{4}{{e}^{x}}^{ln(x)}ln^{4}(x) + 72x{{e}^{x}}^{ln(x)}ln({e}^{x}) + 96x{{e}^{x}}^{ln(x)}ln(x) + 12x^{2}{{e}^{x}}^{ln(x)} + 104x{{e}^{x}}^{ln(x)} + 50{{e}^{x}}^{ln(x)}ln({e}^{x}) + 24{{e}^{x}}^{ln(x)}\\ \end{split}\end{equation} \]

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