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。
\[ \begin{equation}\begin{split}【1/1】求函数{(1 + {x}^{x})}^{\frac{1}{2}} + cos(x) - 2 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = ({x}^{x} + 1)^{\frac{1}{2}} + cos(x) - 2\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( ({x}^{x} + 1)^{\frac{1}{2}} + cos(x) - 2\right)}{dx}\\=&(\frac{\frac{1}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{1}{2}}}) + -sin(x) + 0\\=&\frac{{x}^{x}ln(x)}{2({x}^{x} + 1)^{\frac{1}{2}}} + \frac{{x}^{x}}{2({x}^{x} + 1)^{\frac{1}{2}}} - sin(x)\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{{x}^{x}ln(x)}{2({x}^{x} + 1)^{\frac{1}{2}}} + \frac{{x}^{x}}{2({x}^{x} + 1)^{\frac{1}{2}}} - sin(x)\right)}{dx}\\=&\frac{(\frac{\frac{-1}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{3}{2}}}){x}^{x}ln(x)}{2} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(x)}{2({x}^{x} + 1)^{\frac{1}{2}}} + \frac{{x}^{x}}{2({x}^{x} + 1)^{\frac{1}{2}}(x)} + \frac{(\frac{\frac{-1}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{3}{2}}}){x}^{x}}{2} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{2({x}^{x} + 1)^{\frac{1}{2}}} - cos(x)\\=&\frac{-{x}^{(2x)}ln^{2}(x)}{4({x}^{x} + 1)^{\frac{3}{2}}} + \frac{{x}^{x}ln^{2}(x)}{2({x}^{x} + 1)^{\frac{1}{2}}} - \frac{{x}^{(2x)}ln(x)}{2({x}^{x} + 1)^{\frac{3}{2}}} + \frac{{x}^{x}ln(x)}{({x}^{x} + 1)^{\frac{1}{2}}} + \frac{{x}^{x}}{2({x}^{x} + 1)^{\frac{1}{2}}x} - \frac{{x}^{(2x)}}{4({x}^{x} + 1)^{\frac{3}{2}}} + \frac{{x}^{x}}{2({x}^{x} + 1)^{\frac{1}{2}}} - cos(x)\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-{x}^{(2x)}ln^{2}(x)}{4({x}^{x} + 1)^{\frac{3}{2}}} + \frac{{x}^{x}ln^{2}(x)}{2({x}^{x} + 1)^{\frac{1}{2}}} - \frac{{x}^{(2x)}ln(x)}{2({x}^{x} + 1)^{\frac{3}{2}}} + \frac{{x}^{x}ln(x)}{({x}^{x} + 1)^{\frac{1}{2}}} + \frac{{x}^{x}}{2({x}^{x} + 1)^{\frac{1}{2}}x} - \frac{{x}^{(2x)}}{4({x}^{x} + 1)^{\frac{3}{2}}} + \frac{{x}^{x}}{2({x}^{x} + 1)^{\frac{1}{2}}} - cos(x)\right)}{dx}\\=&\frac{-(\frac{\frac{-3}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{5}{2}}}){x}^{(2x)}ln^{2}(x)}{4} - \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln^{2}(x)}{4({x}^{x} + 1)^{\frac{3}{2}}} - \frac{{x}^{(2x)}*2ln(x)}{4({x}^{x} + 1)^{\frac{3}{2}}(x)} + \frac{(\frac{\frac{-1}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{3}{2}}}){x}^{x}ln^{2}(x)}{2} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln^{2}(x)}{2({x}^{x} + 1)^{\frac{1}{2}}} + \frac{{x}^{x}*2ln(x)}{2({x}^{x} + 1)^{\frac{1}{2}}(x)} - \frac{(\frac{\frac{-3}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{5}{2}}}){x}^{(2x)}ln(x)}{2} - \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln(x)}{2({x}^{x} + 1)^{\frac{3}{2}}} - \frac{{x}^{(2x)}}{2({x}^{x} + 1)^{\frac{3}{2}}(x)} + (\frac{\frac{-1}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{3}{2}}}){x}^{x}ln(x) + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(x)}{({x}^{x} + 1)^{\frac{1}{2}}} + \frac{{x}^{x}}{({x}^{x} + 1)^{\frac{1}{2}}(x)} + \frac{(\frac{\frac{-1}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{3}{2}}}){x}^{x}}{2x} + \frac{-{x}^{x}}{2({x}^{x} + 1)^{\frac{1}{2}}x^{2}} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{2({x}^{x} + 1)^{\frac{1}{2}}x} - \frac{(\frac{\frac{-3}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{5}{2}}}){x}^{(2x)}}{4} - \frac{({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))}{4({x}^{x} + 1)^{\frac{3}{2}}} + \frac{(\frac{\frac{-1}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{3}{2}}}){x}^{x}}{2} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{2({x}^{x} + 1)^{\frac{1}{2}}} - -sin(x)\\=&\frac{3{x}^{(3x)}ln^{3}(x)}{8({x}^{x} + 1)^{\frac{5}{2}}} - \frac{3{x}^{(2x)}ln^{3}(x)}{4({x}^{x} + 1)^{\frac{3}{2}}} + \frac{{x}^{x}ln^{3}(x)}{2({x}^{x} + 1)^{\frac{1}{2}}} - \frac{3{x}^{(2x)}ln(x)}{4({x}^{x} + 1)^{\frac{3}{2}}x} + \frac{9{x}^{(3x)}ln^{2}(x)}{8({x}^{x} + 1)^{\frac{5}{2}}} - \frac{9{x}^{(2x)}ln^{2}(x)}{4({x}^{x} + 1)^{\frac{3}{2}}} + \frac{3{x}^{x}ln^{2}(x)}{2({x}^{x} + 1)^{\frac{1}{2}}} + \frac{3{x}^{x}ln(x)}{2({x}^{x} + 1)^{\frac{1}{2}}x} + \frac{9{x}^{(3x)}ln(x)}{8({x}^{x} + 1)^{\frac{5}{2}}} - \frac{9{x}^{(2x)}ln(x)}{4({x}^{x} + 1)^{\frac{3}{2}}} + \frac{3{x}^{x}ln(x)}{2({x}^{x} + 1)^{\frac{1}{2}}} + \frac{3{x}^{x}}{2({x}^{x} + 1)^{\frac{1}{2}}x} - \frac{3{x}^{(2x)}}{4({x}^{x} + 1)^{\frac{3}{2}}x} - \frac{{x}^{x}}{2({x}^{x} + 1)^{\frac{1}{2}}x^{2}} + \frac{3{x}^{(3x)}}{8({x}^{x} + 1)^{\frac{5}{2}}} - \frac{3{x}^{(2x)}}{4({x}^{x} + 1)^{\frac{3}{2}}} + \frac{{x}^{x}}{2({x}^{x} + 1)^{\frac{1}{2}}} + sin(x)\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{3{x}^{(3x)}ln^{3}(x)}{8({x}^{x} + 1)^{\frac{5}{2}}} - \frac{3{x}^{(2x)}ln^{3}(x)}{4({x}^{x} + 1)^{\frac{3}{2}}} + \frac{{x}^{x}ln^{3}(x)}{2({x}^{x} + 1)^{\frac{1}{2}}} - \frac{3{x}^{(2x)}ln(x)}{4({x}^{x} + 1)^{\frac{3}{2}}x} + \frac{9{x}^{(3x)}ln^{2}(x)}{8({x}^{x} + 1)^{\frac{5}{2}}} - \frac{9{x}^{(2x)}ln^{2}(x)}{4({x}^{x} + 1)^{\frac{3}{2}}} + \frac{3{x}^{x}ln^{2}(x)}{2({x}^{x} + 1)^{\frac{1}{2}}} + \frac{3{x}^{x}ln(x)}{2({x}^{x} + 1)^{\frac{1}{2}}x} + \frac{9{x}^{(3x)}ln(x)}{8({x}^{x} + 1)^{\frac{5}{2}}} - \frac{9{x}^{(2x)}ln(x)}{4({x}^{x} + 1)^{\frac{3}{2}}} + \frac{3{x}^{x}ln(x)}{2({x}^{x} + 1)^{\frac{1}{2}}} + \frac{3{x}^{x}}{2({x}^{x} + 1)^{\frac{1}{2}}x} - \frac{3{x}^{(2x)}}{4({x}^{x} + 1)^{\frac{3}{2}}x} - \frac{{x}^{x}}{2({x}^{x} + 1)^{\frac{1}{2}}x^{2}} + \frac{3{x}^{(3x)}}{8({x}^{x} + 1)^{\frac{5}{2}}} - \frac{3{x}^{(2x)}}{4({x}^{x} + 1)^{\frac{3}{2}}} + \frac{{x}^{x}}{2({x}^{x} + 1)^{\frac{1}{2}}} + sin(x)\right)}{dx}\\=&\frac{3(\frac{\frac{-5}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{7}{2}}}){x}^{(3x)}ln^{3}(x)}{8} + \frac{3({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)}))ln^{3}(x)}{8({x}^{x} + 1)^{\frac{5}{2}}} + \frac{3{x}^{(3x)}*3ln^{2}(x)}{8({x}^{x} + 1)^{\frac{5}{2}}(x)} - \frac{3(\frac{\frac{-3}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{5}{2}}}){x}^{(2x)}ln^{3}(x)}{4} - \frac{3({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln^{3}(x)}{4({x}^{x} + 1)^{\frac{3}{2}}} - \frac{3{x}^{(2x)}*3ln^{2}(x)}{4({x}^{x} + 1)^{\frac{3}{2}}(x)} + \frac{(\frac{\frac{-1}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{3}{2}}}){x}^{x}ln^{3}(x)}{2} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln^{3}(x)}{2({x}^{x} + 1)^{\frac{1}{2}}} + \frac{{x}^{x}*3ln^{2}(x)}{2({x}^{x} + 1)^{\frac{1}{2}}(x)} - \frac{3(\frac{\frac{-3}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{5}{2}}}){x}^{(2x)}ln(x)}{4x} - \frac{3*-{x}^{(2x)}ln(x)}{4({x}^{x} + 1)^{\frac{3}{2}}x^{2}} - \frac{3({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln(x)}{4({x}^{x} + 1)^{\frac{3}{2}}x} - \frac{3{x}^{(2x)}}{4({x}^{x} + 1)^{\frac{3}{2}}x(x)} + \frac{9(\frac{\frac{-5}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{7}{2}}}){x}^{(3x)}ln^{2}(x)}{8} + \frac{9({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)}))ln^{2}(x)}{8({x}^{x} + 1)^{\frac{5}{2}}} + \frac{9{x}^{(3x)}*2ln(x)}{8({x}^{x} + 1)^{\frac{5}{2}}(x)} - \frac{9(\frac{\frac{-3}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{5}{2}}}){x}^{(2x)}ln^{2}(x)}{4} - \frac{9({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln^{2}(x)}{4({x}^{x} + 1)^{\frac{3}{2}}} - \frac{9{x}^{(2x)}*2ln(x)}{4({x}^{x} + 1)^{\frac{3}{2}}(x)} + \frac{3(\frac{\frac{-1}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{3}{2}}}){x}^{x}ln^{2}(x)}{2} + \frac{3({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln^{2}(x)}{2({x}^{x} + 1)^{\frac{1}{2}}} + \frac{3{x}^{x}*2ln(x)}{2({x}^{x} + 1)^{\frac{1}{2}}(x)} + \frac{3(\frac{\frac{-1}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{3}{2}}}){x}^{x}ln(x)}{2x} + \frac{3*-{x}^{x}ln(x)}{2({x}^{x} + 1)^{\frac{1}{2}}x^{2}} + \frac{3({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(x)}{2({x}^{x} + 1)^{\frac{1}{2}}x} + \frac{3{x}^{x}}{2({x}^{x} + 1)^{\frac{1}{2}}x(x)} + \frac{9(\frac{\frac{-5}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{7}{2}}}){x}^{(3x)}ln(x)}{8} + \frac{9({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)}))ln(x)}{8({x}^{x} + 1)^{\frac{5}{2}}} + \frac{9{x}^{(3x)}}{8({x}^{x} + 1)^{\frac{5}{2}}(x)} - \frac{9(\frac{\frac{-3}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{5}{2}}}){x}^{(2x)}ln(x)}{4} - \frac{9({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))ln(x)}{4({x}^{x} + 1)^{\frac{3}{2}}} - \frac{9{x}^{(2x)}}{4({x}^{x} + 1)^{\frac{3}{2}}(x)} + \frac{3(\frac{\frac{-1}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{3}{2}}}){x}^{x}ln(x)}{2} + \frac{3({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))ln(x)}{2({x}^{x} + 1)^{\frac{1}{2}}} + \frac{3{x}^{x}}{2({x}^{x} + 1)^{\frac{1}{2}}(x)} + \frac{3(\frac{\frac{-1}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{3}{2}}}){x}^{x}}{2x} + \frac{3*-{x}^{x}}{2({x}^{x} + 1)^{\frac{1}{2}}x^{2}} + \frac{3({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{2({x}^{x} + 1)^{\frac{1}{2}}x} - \frac{3(\frac{\frac{-3}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{5}{2}}}){x}^{(2x)}}{4x} - \frac{3*-{x}^{(2x)}}{4({x}^{x} + 1)^{\frac{3}{2}}x^{2}} - \frac{3({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))}{4({x}^{x} + 1)^{\frac{3}{2}}x} - \frac{(\frac{\frac{-1}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{3}{2}}}){x}^{x}}{2x^{2}} - \frac{-2{x}^{x}}{2({x}^{x} + 1)^{\frac{1}{2}}x^{3}} - \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{2({x}^{x} + 1)^{\frac{1}{2}}x^{2}} + \frac{3(\frac{\frac{-5}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{7}{2}}}){x}^{(3x)}}{8} + \frac{3({x}^{(3x)}((3)ln(x) + \frac{(3x)(1)}{(x)}))}{8({x}^{x} + 1)^{\frac{5}{2}}} - \frac{3(\frac{\frac{-3}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{5}{2}}}){x}^{(2x)}}{4} - \frac{3({x}^{(2x)}((2)ln(x) + \frac{(2x)(1)}{(x)}))}{4({x}^{x} + 1)^{\frac{3}{2}}} + \frac{(\frac{\frac{-1}{2}(({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)})) + 0)}{({x}^{x} + 1)^{\frac{3}{2}}}){x}^{x}}{2} + \frac{({x}^{x}((1)ln(x) + \frac{(x)(1)}{(x)}))}{2({x}^{x} + 1)^{\frac{1}{2}}} + cos(x)\\=&\frac{-15{x}^{(4x)}ln^{4}(x)}{16({x}^{x} + 1)^{\frac{7}{2}}} + \frac{9{x}^{(3x)}ln^{4}(x)}{4({x}^{x} + 1)^{\frac{5}{2}}} - \frac{7{x}^{(2x)}ln^{4}(x)}{4({x}^{x} + 1)^{\frac{3}{2}}} + \frac{9{x}^{(3x)}ln^{2}(x)}{4({x}^{x} + 1)^{\frac{5}{2}}x} + \frac{{x}^{x}ln^{4}(x)}{2({x}^{x} + 1)^{\frac{1}{2}}} - \frac{15{x}^{(4x)}ln^{3}(x)}{4({x}^{x} + 1)^{\frac{7}{2}}} + \frac{9{x}^{(3x)}ln^{3}(x)}{({x}^{x} + 1)^{\frac{5}{2}}} - \frac{9{x}^{(2x)}ln^{2}(x)}{2({x}^{x} + 1)^{\frac{3}{2}}x} - \frac{7{x}^{(2x)}ln^{3}(x)}{({x}^{x} + 1)^{\frac{3}{2}}} + \frac{2{x}^{x}ln^{3}(x)}{({x}^{x} + 1)^{\frac{1}{2}}} + \frac{3{x}^{x}ln^{2}(x)}{({x}^{x} + 1)^{\frac{1}{2}}x} + \frac{{x}^{(2x)}ln(x)}{({x}^{x} + 1)^{\frac{3}{2}}x^{2}} - \frac{45{x}^{(4x)}ln^{2}(x)}{8({x}^{x} + 1)^{\frac{7}{2}}} + \frac{27{x}^{(3x)}ln^{2}(x)}{2({x}^{x} + 1)^{\frac{5}{2}}} - \frac{21{x}^{(2x)}ln^{2}(x)}{2({x}^{x} + 1)^{\frac{3}{2}}} + \frac{9{x}^{(3x)}ln(x)}{2({x}^{x} + 1)^{\frac{5}{2}}x} + \frac{3{x}^{x}ln^{2}(x)}{({x}^{x} + 1)^{\frac{1}{2}}} - \frac{9{x}^{(2x)}ln(x)}{({x}^{x} + 1)^{\frac{3}{2}}x} + \frac{6{x}^{x}ln(x)}{({x}^{x} + 1)^{\frac{1}{2}}x} - \frac{2{x}^{x}ln(x)}{({x}^{x} + 1)^{\frac{1}{2}}x^{2}} - \frac{15{x}^{(4x)}ln(x)}{4({x}^{x} + 1)^{\frac{7}{2}}} + \frac{9{x}^{(3x)}ln(x)}{({x}^{x} + 1)^{\frac{5}{2}}} - \frac{7{x}^{(2x)}ln(x)}{({x}^{x} + 1)^{\frac{3}{2}}} - \frac{9{x}^{(2x)}}{2({x}^{x} + 1)^{\frac{3}{2}}x} + \frac{2{x}^{x}ln(x)}{({x}^{x} + 1)^{\frac{1}{2}}} + \frac{3{x}^{x}}{({x}^{x} + 1)^{\frac{1}{2}}x} + \frac{9{x}^{(3x)}}{4({x}^{x} + 1)^{\frac{5}{2}}x} + \frac{{x}^{(2x)}}{4({x}^{x} + 1)^{\frac{3}{2}}x^{2}} + \frac{{x}^{x}}{({x}^{x} + 1)^{\frac{1}{2}}x^{3}} - \frac{{x}^{x}}{2({x}^{x} + 1)^{\frac{1}{2}}x^{2}} + \frac{9{x}^{(3x)}}{4({x}^{x} + 1)^{\frac{5}{2}}} - \frac{7{x}^{(2x)}}{4({x}^{x} + 1)^{\frac{3}{2}}} - \frac{15{x}^{(4x)}}{16({x}^{x} + 1)^{\frac{7}{2}}} + \frac{{x}^{x}}{2({x}^{x} + 1)^{\frac{1}{2}}} + cos(x)\\ \end{split}\end{equation} \]

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