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。
$$\begin{equation}\begin{split}【1/1】求函数{({x}^{2} - 1)}^{n} 关于 x 的 4 阶导数：\\\end{split}\end{equation}$$ $$\begin{equation}\begin{split}\\解:&\\ &原函数 = (x^{2} - 1)^{n}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( (x^{2} - 1)^{n}\right)}{dx}\\=&((x^{2} - 1)^{n}((0)ln(x^{2} - 1) + \frac{(n)(2x + 0)}{(x^{2} - 1)}))\\=&\frac{2nx(x^{2} - 1)^{n}}{(x^{2} - 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{2nx(x^{2} - 1)^{n}}{(x^{2} - 1)}\right)}{dx}\\=&2(\frac{-(2x + 0)}{(x^{2} - 1)^{2}})nx(x^{2} - 1)^{n} + \frac{2n(x^{2} - 1)^{n}}{(x^{2} - 1)} + \frac{2nx((x^{2} - 1)^{n}((0)ln(x^{2} - 1) + \frac{(n)(2x + 0)}{(x^{2} - 1)}))}{(x^{2} - 1)}\\=&\frac{-4nx^{2}(x^{2} - 1)^{n}}{(x^{2} - 1)^{2}} + \frac{2n(x^{2} - 1)^{n}}{(x^{2} - 1)} + \frac{4n^{2}x^{2}(x^{2} - 1)^{n}}{(x^{2} - 1)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-4nx^{2}(x^{2} - 1)^{n}}{(x^{2} - 1)^{2}} + \frac{2n(x^{2} - 1)^{n}}{(x^{2} - 1)} + \frac{4n^{2}x^{2}(x^{2} - 1)^{n}}{(x^{2} - 1)^{2}}\right)}{dx}\\=&-4(\frac{-2(2x + 0)}{(x^{2} - 1)^{3}})nx^{2}(x^{2} - 1)^{n} - \frac{4n*2x(x^{2} - 1)^{n}}{(x^{2} - 1)^{2}} - \frac{4nx^{2}((x^{2} - 1)^{n}((0)ln(x^{2} - 1) + \frac{(n)(2x + 0)}{(x^{2} - 1)}))}{(x^{2} - 1)^{2}} + 2(\frac{-(2x + 0)}{(x^{2} - 1)^{2}})n(x^{2} - 1)^{n} + \frac{2n((x^{2} - 1)^{n}((0)ln(x^{2} - 1) + \frac{(n)(2x + 0)}{(x^{2} - 1)}))}{(x^{2} - 1)} + 4(\frac{-2(2x + 0)}{(x^{2} - 1)^{3}})n^{2}x^{2}(x^{2} - 1)^{n} + \frac{4n^{2}*2x(x^{2} - 1)^{n}}{(x^{2} - 1)^{2}} + \frac{4n^{2}x^{2}((x^{2} - 1)^{n}((0)ln(x^{2} - 1) + \frac{(n)(2x + 0)}{(x^{2} - 1)}))}{(x^{2} - 1)^{2}}\\=&\frac{16nx^{3}(x^{2} - 1)^{n}}{(x^{2} - 1)^{3}} - \frac{12nx(x^{2} - 1)^{n}}{(x^{2} - 1)^{2}} - \frac{24n^{2}x^{3}(x^{2} - 1)^{n}}{(x^{2} - 1)^{3}} + \frac{12n^{2}x(x^{2} - 1)^{n}}{(x^{2} - 1)^{2}} + \frac{8n^{3}x^{3}(x^{2} - 1)^{n}}{(x^{2} - 1)^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{16nx^{3}(x^{2} - 1)^{n}}{(x^{2} - 1)^{3}} - \frac{12nx(x^{2} - 1)^{n}}{(x^{2} - 1)^{2}} - \frac{24n^{2}x^{3}(x^{2} - 1)^{n}}{(x^{2} - 1)^{3}} + \frac{12n^{2}x(x^{2} - 1)^{n}}{(x^{2} - 1)^{2}} + \frac{8n^{3}x^{3}(x^{2} - 1)^{n}}{(x^{2} - 1)^{3}}\right)}{dx}\\=&16(\frac{-3(2x + 0)}{(x^{2} - 1)^{4}})nx^{3}(x^{2} - 1)^{n} + \frac{16n*3x^{2}(x^{2} - 1)^{n}}{(x^{2} - 1)^{3}} + \frac{16nx^{3}((x^{2} - 1)^{n}((0)ln(x^{2} - 1) + \frac{(n)(2x + 0)}{(x^{2} - 1)}))}{(x^{2} - 1)^{3}} - 12(\frac{-2(2x + 0)}{(x^{2} - 1)^{3}})nx(x^{2} - 1)^{n} - \frac{12n(x^{2} - 1)^{n}}{(x^{2} - 1)^{2}} - \frac{12nx((x^{2} - 1)^{n}((0)ln(x^{2} - 1) + \frac{(n)(2x + 0)}{(x^{2} - 1)}))}{(x^{2} - 1)^{2}} - 24(\frac{-3(2x + 0)}{(x^{2} - 1)^{4}})n^{2}x^{3}(x^{2} - 1)^{n} - \frac{24n^{2}*3x^{2}(x^{2} - 1)^{n}}{(x^{2} - 1)^{3}} - \frac{24n^{2}x^{3}((x^{2} - 1)^{n}((0)ln(x^{2} - 1) + \frac{(n)(2x + 0)}{(x^{2} - 1)}))}{(x^{2} - 1)^{3}} + 12(\frac{-2(2x + 0)}{(x^{2} - 1)^{3}})n^{2}x(x^{2} - 1)^{n} + \frac{12n^{2}(x^{2} - 1)^{n}}{(x^{2} - 1)^{2}} + \frac{12n^{2}x((x^{2} - 1)^{n}((0)ln(x^{2} - 1) + \frac{(n)(2x + 0)}{(x^{2} - 1)}))}{(x^{2} - 1)^{2}} + 8(\frac{-3(2x + 0)}{(x^{2} - 1)^{4}})n^{3}x^{3}(x^{2} - 1)^{n} + \frac{8n^{3}*3x^{2}(x^{2} - 1)^{n}}{(x^{2} - 1)^{3}} + \frac{8n^{3}x^{3}((x^{2} - 1)^{n}((0)ln(x^{2} - 1) + \frac{(n)(2x + 0)}{(x^{2} - 1)}))}{(x^{2} - 1)^{3}}\\=&\frac{-96nx^{4}(x^{2} - 1)^{n}}{(x^{2} - 1)^{4}} + \frac{96nx^{2}(x^{2} - 1)^{n}}{(x^{2} - 1)^{3}} + \frac{176n^{2}x^{4}(x^{2} - 1)^{n}}{(x^{2} - 1)^{4}} - \frac{12n(x^{2} - 1)^{n}}{(x^{2} - 1)^{2}} - \frac{144n^{2}x^{2}(x^{2} - 1)^{n}}{(x^{2} - 1)^{3}} - \frac{96n^{3}x^{4}(x^{2} - 1)^{n}}{(x^{2} - 1)^{4}} + \frac{12n^{2}(x^{2} - 1)^{n}}{(x^{2} - 1)^{2}} + \frac{48n^{3}x^{2}(x^{2} - 1)^{n}}{(x^{2} - 1)^{3}} + \frac{16n^{4}x^{4}(x^{2} - 1)^{n}}{(x^{2} - 1)^{4}}\\ \end{split}\end{equation}$$

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