求导函数：
输入一个原函数（即需要求导的函数），然后设置需要求导的变量和求导的阶数，点击“下一步”按钮，即可获得该函数相应阶数的导函数。
注意，输入的函数支持数学函数和其它常量。
输入一个原函数（即需要求导的函数），然后设置需要求导的变量和求导的阶数，点击“下一步”按钮，即可获得该函数相应阶数的导函数。
注意，输入的函数支持数学函数和其它常量。
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。
\[ \begin{equation}\begin{split}【1/1】求函数{({x}^{2} + 1)}^{x} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = (x^{2} + 1)^{x}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( (x^{2} + 1)^{x}\right)}{dx}\\=&((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))\\=&(x^{2} + 1)^{x}ln(x^{2} + 1) + \frac{2x^{2}(x^{2} + 1)^{x}}{(x^{2} + 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( (x^{2} + 1)^{x}ln(x^{2} + 1) + \frac{2x^{2}(x^{2} + 1)^{x}}{(x^{2} + 1)}\right)}{dx}\\=&((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))ln(x^{2} + 1) + \frac{(x^{2} + 1)^{x}(2x + 0)}{(x^{2} + 1)} + 2(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x^{2}(x^{2} + 1)^{x} + \frac{2*2x(x^{2} + 1)^{x}}{(x^{2} + 1)} + \frac{2x^{2}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))}{(x^{2} + 1)}\\=&(x^{2} + 1)^{x}ln^{2}(x^{2} + 1) + \frac{4x^{2}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)} + \frac{6x(x^{2} + 1)^{x}}{(x^{2} + 1)} - \frac{4x^{3}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} + \frac{4x^{4}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( (x^{2} + 1)^{x}ln^{2}(x^{2} + 1) + \frac{4x^{2}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)} + \frac{6x(x^{2} + 1)^{x}}{(x^{2} + 1)} - \frac{4x^{3}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} + \frac{4x^{4}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}}\right)}{dx}\\=&((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))ln^{2}(x^{2} + 1) + \frac{(x^{2} + 1)^{x}*2ln(x^{2} + 1)(2x + 0)}{(x^{2} + 1)} + 4(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x^{2}(x^{2} + 1)^{x}ln(x^{2} + 1) + \frac{4*2x(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)} + \frac{4x^{2}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))ln(x^{2} + 1)}{(x^{2} + 1)} + \frac{4x^{2}(x^{2} + 1)^{x}(2x + 0)}{(x^{2} + 1)(x^{2} + 1)} + 6(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x(x^{2} + 1)^{x} + \frac{6(x^{2} + 1)^{x}}{(x^{2} + 1)} + \frac{6x((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))}{(x^{2} + 1)} - 4(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{3}(x^{2} + 1)^{x} - \frac{4*3x^{2}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} - \frac{4x^{3}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))}{(x^{2} + 1)^{2}} + 4(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{4}(x^{2} + 1)^{x} + \frac{4*4x^{3}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} + \frac{4x^{4}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))}{(x^{2} + 1)^{2}}\\=&(x^{2} + 1)^{x}ln^{3}(x^{2} + 1) + \frac{6x^{2}(x^{2} + 1)^{x}ln^{2}(x^{2} + 1)}{(x^{2} + 1)} + \frac{18x(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)} - \frac{12x^{3}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)^{2}} + \frac{12x^{4}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)^{2}} + \frac{36x^{3}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} - \frac{24x^{2}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} + \frac{6(x^{2} + 1)^{x}}{(x^{2} + 1)} + \frac{16x^{4}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}} - \frac{24x^{5}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}} + \frac{8x^{6}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( (x^{2} + 1)^{x}ln^{3}(x^{2} + 1) + \frac{6x^{2}(x^{2} + 1)^{x}ln^{2}(x^{2} + 1)}{(x^{2} + 1)} + \frac{18x(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)} - \frac{12x^{3}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)^{2}} + \frac{12x^{4}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)^{2}} + \frac{36x^{3}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} - \frac{24x^{2}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} + \frac{6(x^{2} + 1)^{x}}{(x^{2} + 1)} + \frac{16x^{4}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}} - \frac{24x^{5}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}} + \frac{8x^{6}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}}\right)}{dx}\\=&((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))ln^{3}(x^{2} + 1) + \frac{(x^{2} + 1)^{x}*3ln^{2}(x^{2} + 1)(2x + 0)}{(x^{2} + 1)} + 6(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x^{2}(x^{2} + 1)^{x}ln^{2}(x^{2} + 1) + \frac{6*2x(x^{2} + 1)^{x}ln^{2}(x^{2} + 1)}{(x^{2} + 1)} + \frac{6x^{2}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))ln^{2}(x^{2} + 1)}{(x^{2} + 1)} + \frac{6x^{2}(x^{2} + 1)^{x}*2ln(x^{2} + 1)(2x + 0)}{(x^{2} + 1)(x^{2} + 1)} + 18(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x(x^{2} + 1)^{x}ln(x^{2} + 1) + \frac{18(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)} + \frac{18x((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))ln(x^{2} + 1)}{(x^{2} + 1)} + \frac{18x(x^{2} + 1)^{x}(2x + 0)}{(x^{2} + 1)(x^{2} + 1)} - 12(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{3}(x^{2} + 1)^{x}ln(x^{2} + 1) - \frac{12*3x^{2}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)^{2}} - \frac{12x^{3}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))ln(x^{2} + 1)}{(x^{2} + 1)^{2}} - \frac{12x^{3}(x^{2} + 1)^{x}(2x + 0)}{(x^{2} + 1)^{2}(x^{2} + 1)} + 12(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{4}(x^{2} + 1)^{x}ln(x^{2} + 1) + \frac{12*4x^{3}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)^{2}} + \frac{12x^{4}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))ln(x^{2} + 1)}{(x^{2} + 1)^{2}} + \frac{12x^{4}(x^{2} + 1)^{x}(2x + 0)}{(x^{2} + 1)^{2}(x^{2} + 1)} + 36(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{3}(x^{2} + 1)^{x} + \frac{36*3x^{2}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} + \frac{36x^{3}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))}{(x^{2} + 1)^{2}} - 24(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{2}(x^{2} + 1)^{x} - \frac{24*2x(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} - \frac{24x^{2}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))}{(x^{2} + 1)^{2}} + 6(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})(x^{2} + 1)^{x} + \frac{6((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))}{(x^{2} + 1)} + 16(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{4}(x^{2} + 1)^{x} + \frac{16*4x^{3}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}} + \frac{16x^{4}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))}{(x^{2} + 1)^{3}} - 24(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{5}(x^{2} + 1)^{x} - \frac{24*5x^{4}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}} - \frac{24x^{5}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))}{(x^{2} + 1)^{3}} + 8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{6}(x^{2} + 1)^{x} + \frac{8*6x^{5}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}} + \frac{8x^{6}((x^{2} + 1)^{x}((1)ln(x^{2} + 1) + \frac{(x)(2x + 0)}{(x^{2} + 1)}))}{(x^{2} + 1)^{3}}\\=&(x^{2} + 1)^{x}ln^{4}(x^{2} + 1) + \frac{8x^{2}(x^{2} + 1)^{x}ln^{3}(x^{2} + 1)}{(x^{2} + 1)} + \frac{36x(x^{2} + 1)^{x}ln^{2}(x^{2} + 1)}{(x^{2} + 1)} - \frac{24x^{3}(x^{2} + 1)^{x}ln^{2}(x^{2} + 1)}{(x^{2} + 1)^{2}} + \frac{24x^{4}(x^{2} + 1)^{x}ln^{2}(x^{2} + 1)}{(x^{2} + 1)^{2}} + \frac{144x^{3}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)^{2}} - \frac{96x^{2}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)^{2}} + \frac{24(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)} + \frac{64x^{4}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)^{3}} - \frac{96x^{5}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)^{3}} + \frac{32x^{6}(x^{2} + 1)^{x}ln(x^{2} + 1)}{(x^{2} + 1)^{3}} - \frac{336x^{4}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}} + \frac{156x^{2}(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} + \frac{144x^{5}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}} + \frac{160x^{3}(x^{2} + 1)^{x}}{(x^{2} + 1)^{3}} - \frac{60x(x^{2} + 1)^{x}}{(x^{2} + 1)^{2}} - \frac{96x^{5}(x^{2} + 1)^{x}}{(x^{2} + 1)^{4}} + \frac{176x^{6}(x^{2} + 1)^{x}}{(x^{2} + 1)^{4}} - \frac{96x^{7}(x^{2} + 1)^{x}}{(x^{2} + 1)^{4}} + \frac{16x^{8}(x^{2} + 1)^{x}}{(x^{2} + 1)^{4}}\\ \end{split}\end{equation} \]
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