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。
\[ \begin{equation}\begin{split}【1/1】求函数{e}^{({x}^{x} - 1)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( {e}^{({x}^{x} - 1)}\right)}{dx}\\=&({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)}))\\=&{x}^{x}{e}^{({x}^{x} - 1)}ln(x) + {x}^{x}{e}^{({x}^{x} - 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( {x}^{x}{e}^{({x}^{x} - 1)}ln(x) + {x}^{x}{e}^{({x}^{x} - 1)}\right)}{dx}\\=&({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){e}^{({x}^{x} - 1)}ln(x) + {x}^{x}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)}))ln(x) + \frac{{x}^{x}{e}^{({x}^{x} - 1)}}{(x)} + ({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){e}^{({x}^{x} - 1)} + {x}^{x}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)}))\\=&{x}^{x}{e}^{({x}^{x} - 1)}ln^{2}(x) + 2{x}^{x}{e}^{({x}^{x} - 1)}ln(x) + {x}^{(2(x))}{e}^{({x}^{x} - 1)}ln^{2}(x) + 2{x}^{(2(x))}{e}^{({x}^{x} - 1)}ln(x) + \frac{{e}^{({x}^{x} - 1)}{x}^{x}}{x} + {x}^{x}{e}^{({x}^{x} - 1)} + {x}^{(2(x))}{e}^{({x}^{x} - 1)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( {x}^{x}{e}^{({x}^{x} - 1)}ln^{2}(x) + 2{x}^{x}{e}^{({x}^{x} - 1)}ln(x) + {x}^{(2(x))}{e}^{({x}^{x} - 1)}ln^{2}(x) + 2{x}^{(2(x))}{e}^{({x}^{x} - 1)}ln(x) + \frac{{e}^{({x}^{x} - 1)}{x}^{x}}{x} + {x}^{x}{e}^{({x}^{x} - 1)} + {x}^{(2(x))}{e}^{({x}^{x} - 1)}\right)}{dx}\\=&({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){e}^{({x}^{x} - 1)}ln^{2}(x) + {x}^{x}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)}))ln^{2}(x) + \frac{{x}^{x}{e}^{({x}^{x} - 1)}*2ln(x)}{(x)} + 2({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){e}^{({x}^{x} - 1)}ln(x) + 2{x}^{x}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)}))ln(x) + \frac{2{x}^{x}{e}^{({x}^{x} - 1)}}{(x)} + ({x}^{(2(x))}((2(1))ln(x) + \frac{(2(x))(1)}{(x)})){e}^{({x}^{x} - 1)}ln^{2}(x) + {x}^{(2(x))}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)}))ln^{2}(x) + \frac{{x}^{(2(x))}{e}^{({x}^{x} - 1)}*2ln(x)}{(x)} + 2({x}^{(2(x))}((2(1))ln(x) + \frac{(2(x))(1)}{(x)})){e}^{({x}^{x} - 1)}ln(x) + 2{x}^{(2(x))}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)}))ln(x) + \frac{2{x}^{(2(x))}{e}^{({x}^{x} - 1)}}{(x)} + \frac{-{e}^{({x}^{x} - 1)}{x}^{x}}{x^{2}} + \frac{({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)})){x}^{x}}{x} + \frac{{e}^{({x}^{x} - 1)}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{x} + ({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){e}^{({x}^{x} - 1)} + {x}^{x}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)})) + ({x}^{(2(x))}((2(1))ln(x) + \frac{(2(x))(1)}{(x)})){e}^{({x}^{x} - 1)} + {x}^{(2(x))}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)}))\\=&{x}^{x}{e}^{({x}^{x} - 1)}ln^{3}(x) + 3{x}^{x}{e}^{({x}^{x} - 1)}ln^{2}(x) + 3{x}^{(2x)}{e}^{({x}^{x} - 1)}ln^{3}(x) + 9{x}^{(2x)}{e}^{({x}^{x} - 1)}ln^{2}(x) + \frac{3{x}^{x}{e}^{({x}^{x} - 1)}ln(x)}{x} + {x}^{(3x)}{e}^{({x}^{x} - 1)}ln^{3}(x) + 3{x}^{(3x)}{e}^{({x}^{x} - 1)}ln^{2}(x) + \frac{3{x}^{(2x)}{e}^{({x}^{x} - 1)}ln(x)}{x} + 3{x}^{x}{e}^{({x}^{x} - 1)}ln(x) + 9{x}^{(2x)}{e}^{({x}^{x} - 1)}ln(x) + 3{x}^{(3x)}{e}^{({x}^{x} - 1)}ln(x) + \frac{2{e}^{({x}^{x} - 1)}{x}^{(2x)}}{x} - \frac{{e}^{({x}^{x} - 1)}{x}^{x}}{x^{2}} + \frac{2{e}^{({x}^{x} - 1)}{x}^{x}}{x} + \frac{{x}^{(2x)}{e}^{({x}^{x} - 1)}}{x} + \frac{{x}^{x}{e}^{({x}^{x} - 1)}}{x} + {x}^{x}{e}^{({x}^{x} - 1)} + 3{x}^{(2x)}{e}^{({x}^{x} - 1)} + {x}^{(3x)}{e}^{({x}^{x} - 1)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( {x}^{x}{e}^{({x}^{x} - 1)}ln^{3}(x) + 3{x}^{x}{e}^{({x}^{x} - 1)}ln^{2}(x) + 3{x}^{(2x)}{e}^{({x}^{x} - 1)}ln^{3}(x) + 9{x}^{(2x)}{e}^{({x}^{x} - 1)}ln^{2}(x) + \frac{3{x}^{x}{e}^{({x}^{x} - 1)}ln(x)}{x} + {x}^{(3x)}{e}^{({x}^{x} - 1)}ln^{3}(x) + 3{x}^{(3x)}{e}^{({x}^{x} - 1)}ln^{2}(x) + \frac{3{x}^{(2x)}{e}^{({x}^{x} - 1)}ln(x)}{x} + 3{x}^{x}{e}^{({x}^{x} - 1)}ln(x) + 9{x}^{(2x)}{e}^{({x}^{x} - 1)}ln(x) + 3{x}^{(3x)}{e}^{({x}^{x} - 1)}ln(x) + \frac{2{e}^{({x}^{x} - 1)}{x}^{(2x)}}{x} - \frac{{e}^{({x}^{x} - 1)}{x}^{x}}{x^{2}} + \frac{2{e}^{({x}^{x} - 1)}{x}^{x}}{x} + \frac{{x}^{(2x)}{e}^{({x}^{x} - 1)}}{x} + \frac{{x}^{x}{e}^{({x}^{x} - 1)}}{x} + {x}^{x}{e}^{({x}^{x} - 1)} + 3{x}^{(2x)}{e}^{({x}^{x} - 1)} + {x}^{(3x)}{e}^{({x}^{x} - 1)}\right)}{dx}\\=&({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){e}^{({x}^{x} - 1)}ln^{3}(x) + {x}^{x}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)}))ln^{3}(x) + \frac{{x}^{x}{e}^{({x}^{x} - 1)}*3ln^{2}(x)}{(x)} + 3({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){e}^{({x}^{x} - 1)}ln^{2}(x) + 3{x}^{x}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)}))ln^{2}(x) + \frac{3{x}^{x}{e}^{({x}^{x} - 1)}*2ln(x)}{(x)} + 3({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){e}^{({x}^{x} - 1)}ln^{3}(x) + 3{x}^{(2x)}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)}))ln^{3}(x) + \frac{3{x}^{(2x)}{e}^{({x}^{x} - 1)}*3ln^{2}(x)}{(x)} + 9({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){e}^{({x}^{x} - 1)}ln^{2}(x) + 9{x}^{(2x)}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)}))ln^{2}(x) + \frac{9{x}^{(2x)}{e}^{({x}^{x} - 1)}*2ln(x)}{(x)} + \frac{3*-{x}^{x}{e}^{({x}^{x} - 1)}ln(x)}{x^{2}} + \frac{3({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){e}^{({x}^{x} - 1)}ln(x)}{x} + \frac{3{x}^{x}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)}))ln(x)}{x} + \frac{3{x}^{x}{e}^{({x}^{x} - 1)}}{x(x)} + ({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){e}^{({x}^{x} - 1)}ln^{3}(x) + {x}^{(3x)}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)}))ln^{3}(x) + \frac{{x}^{(3x)}{e}^{({x}^{x} - 1)}*3ln^{2}(x)}{(x)} + 3({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){e}^{({x}^{x} - 1)}ln^{2}(x) + 3{x}^{(3x)}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)}))ln^{2}(x) + \frac{3{x}^{(3x)}{e}^{({x}^{x} - 1)}*2ln(x)}{(x)} + \frac{3*-{x}^{(2x)}{e}^{({x}^{x} - 1)}ln(x)}{x^{2}} + \frac{3({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){e}^{({x}^{x} - 1)}ln(x)}{x} + \frac{3{x}^{(2x)}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)}))ln(x)}{x} + \frac{3{x}^{(2x)}{e}^{({x}^{x} - 1)}}{x(x)} + 3({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){e}^{({x}^{x} - 1)}ln(x) + 3{x}^{x}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)}))ln(x) + \frac{3{x}^{x}{e}^{({x}^{x} - 1)}}{(x)} + 9({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){e}^{({x}^{x} - 1)}ln(x) + 9{x}^{(2x)}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)}))ln(x) + \frac{9{x}^{(2x)}{e}^{({x}^{x} - 1)}}{(x)} + 3({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){e}^{({x}^{x} - 1)}ln(x) + 3{x}^{(3x)}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)}))ln(x) + \frac{3{x}^{(3x)}{e}^{({x}^{x} - 1)}}{(x)} + \frac{2*-{e}^{({x}^{x} - 1)}{x}^{(2x)}}{x^{2}} + \frac{2({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)})){x}^{(2x)}}{x} + \frac{2{e}^{({x}^{x} - 1)}({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))}{x} - \frac{-2{e}^{({x}^{x} - 1)}{x}^{x}}{x^{3}} - \frac{({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)})){x}^{x}}{x^{2}} - \frac{{e}^{({x}^{x} - 1)}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{x^{2}} + \frac{2*-{e}^{({x}^{x} - 1)}{x}^{x}}{x^{2}} + \frac{2({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)})){x}^{x}}{x} + \frac{2{e}^{({x}^{x} - 1)}({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{x} + \frac{-{x}^{(2x)}{e}^{({x}^{x} - 1)}}{x^{2}} + \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){e}^{({x}^{x} - 1)}}{x} + \frac{{x}^{(2x)}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)}))}{x} + \frac{-{x}^{x}{e}^{({x}^{x} - 1)}}{x^{2}} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){e}^{({x}^{x} - 1)}}{x} + \frac{{x}^{x}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)}))}{x} + ({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})){e}^{({x}^{x} - 1)} + {x}^{x}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)})) + 3({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)})){e}^{({x}^{x} - 1)} + 3{x}^{(2x)}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)})) + ({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)})){e}^{({x}^{x} - 1)} + {x}^{(3x)}({e}^{({x}^{x} - 1)}((({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)ln(e) + \frac{({x}^{x} - 1)(0)}{(e)}))\\=&{x}^{x}{e}^{({x}^{x} - 1)}ln^{4}(x) + 4{x}^{x}{e}^{({x}^{x} - 1)}ln^{3}(x) + 7{x}^{(2x)}{e}^{({x}^{x} - 1)}ln^{4}(x) + 28{x}^{(2x)}{e}^{({x}^{x} - 1)}ln^{3}(x) + \frac{6{x}^{x}{e}^{({x}^{x} - 1)}ln^{2}(x)}{x} + 6{x}^{(3x)}{e}^{({x}^{x} - 1)}ln^{4}(x) + 24{x}^{(3x)}{e}^{({x}^{x} - 1)}ln^{3}(x) + \frac{12{x}^{x}{e}^{({x}^{x} - 1)}ln(x)}{x} + {x}^{(4x)}{e}^{({x}^{x} - 1)}ln^{4}(x) + 4{x}^{(4x)}{e}^{({x}^{x} - 1)}ln^{3}(x) + \frac{18{x}^{(2x)}{e}^{({x}^{x} - 1)}ln^{2}(x)}{x} + 6{x}^{x}{e}^{({x}^{x} - 1)}ln^{2}(x) + 42{x}^{(2x)}{e}^{({x}^{x} - 1)}ln^{2}(x) + 36{x}^{(3x)}{e}^{({x}^{x} - 1)}ln^{2}(x) + \frac{36{x}^{(2x)}{e}^{({x}^{x} - 1)}ln(x)}{x} - \frac{4{x}^{x}{e}^{({x}^{x} - 1)}ln(x)}{x^{2}} + \frac{6{x}^{(3x)}{e}^{({x}^{x} - 1)}ln^{2}(x)}{x} + 6{x}^{(4x)}{e}^{({x}^{x} - 1)}ln^{2}(x) + \frac{12{x}^{(3x)}{e}^{({x}^{x} - 1)}ln(x)}{x} - \frac{4{x}^{(2x)}{e}^{({x}^{x} - 1)}ln(x)}{x^{2}} + \frac{{e}^{({x}^{x} - 1)}{x}^{(2x)}}{x^{2}} + 4{x}^{x}{e}^{({x}^{x} - 1)}ln(x) + 28{x}^{(2x)}{e}^{({x}^{x} - 1)}ln(x) + \frac{3{e}^{({x}^{x} - 1)}{x}^{x}}{x} + 24{x}^{(3x)}{e}^{({x}^{x} - 1)}ln(x) + \frac{9{e}^{({x}^{x} - 1)}{x}^{(2x)}}{x} + 4{x}^{(4x)}{e}^{({x}^{x} - 1)}ln(x) + \frac{3{e}^{({x}^{x} - 1)}{x}^{(3x)}}{x} + \frac{{e}^{({x}^{x} - 1)}{x}^{x}}{x^{2}} + \frac{2{e}^{({x}^{x} - 1)}{x}^{x}}{x^{3}} - \frac{2{x}^{(2x)}{e}^{({x}^{x} - 1)}}{x^{2}} - \frac{2{x}^{x}{e}^{({x}^{x} - 1)}}{x^{2}} + \frac{3{x}^{(3x)}{e}^{({x}^{x} - 1)}}{x} + \frac{9{x}^{(2x)}{e}^{({x}^{x} - 1)}}{x} + \frac{3{x}^{x}{e}^{({x}^{x} - 1)}}{x} + 7{x}^{(2x)}{e}^{({x}^{x} - 1)} + 6{x}^{(3x)}{e}^{({x}^{x} - 1)} + {x}^{x}{e}^{({x}^{x} - 1)} + {x}^{(4x)}{e}^{({x}^{x} - 1)}\\ \end{split}\end{equation} \]

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