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。
\[ \begin{equation}\begin{split}【1/1】求函数{({{1}^{sin({3}^{x})}}^{2})}^{4} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = {1}^{(8sin({3}^{x}))}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( {1}^{(8sin({3}^{x}))}\right)}{dx}\\=&({1}^{(8sin({3}^{x}))}((8cos({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})))ln(1) + \frac{(8sin({3}^{x}))(0)}{(1)}))\\=&8 * {3}^{x}{1}^{(8sin({3}^{x}))}ln(1)ln(3)cos({3}^{x})\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( 8 * {3}^{x}{1}^{(8sin({3}^{x}))}ln(1)ln(3)cos({3}^{x})\right)}{dx}\\=&8({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})){1}^{(8sin({3}^{x}))}ln(1)ln(3)cos({3}^{x}) + 8 * {3}^{x}({1}^{(8sin({3}^{x}))}((8cos({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})))ln(1) + \frac{(8sin({3}^{x}))(0)}{(1)}))ln(1)ln(3)cos({3}^{x}) + \frac{8 * {3}^{x}{1}^{(8sin({3}^{x}))}*0ln(3)cos({3}^{x})}{(1)} + \frac{8 * {3}^{x}{1}^{(8sin({3}^{x}))}ln(1)*0cos({3}^{x})}{(3)} + 8 * {3}^{x}{1}^{(8sin({3}^{x}))}ln(1)ln(3)*-sin({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)}))\\=&8 * {3}^{x}{1}^{(8sin({3}^{x}))}ln^{2}(3)ln(1)cos({3}^{x}) + 64 * {3}^{(2x)}{1}^{(8sin({3}^{x}))}ln^{2}(3)ln^{2}(1)cos^{2}({3}^{x}) - 8 * {1}^{(8sin({3}^{x}))}{3}^{(2x)}ln^{2}(3)ln(1)sin({3}^{x})\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( 8 * {3}^{x}{1}^{(8sin({3}^{x}))}ln^{2}(3)ln(1)cos({3}^{x}) + 64 * {3}^{(2x)}{1}^{(8sin({3}^{x}))}ln^{2}(3)ln^{2}(1)cos^{2}({3}^{x}) - 8 * {1}^{(8sin({3}^{x}))}{3}^{(2x)}ln^{2}(3)ln(1)sin({3}^{x})\right)}{dx}\\=&8({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})){1}^{(8sin({3}^{x}))}ln^{2}(3)ln(1)cos({3}^{x}) + 8 * {3}^{x}({1}^{(8sin({3}^{x}))}((8cos({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})))ln(1) + \frac{(8sin({3}^{x}))(0)}{(1)}))ln^{2}(3)ln(1)cos({3}^{x}) + \frac{8 * {3}^{x}{1}^{(8sin({3}^{x}))}*2ln(3)*0ln(1)cos({3}^{x})}{(3)} + \frac{8 * {3}^{x}{1}^{(8sin({3}^{x}))}ln^{2}(3)*0cos({3}^{x})}{(1)} + 8 * {3}^{x}{1}^{(8sin({3}^{x}))}ln^{2}(3)ln(1)*-sin({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})) + 64({3}^{(2x)}((2)ln(3) + \frac{(2x)(0)}{(3)})){1}^{(8sin({3}^{x}))}ln^{2}(3)ln^{2}(1)cos^{2}({3}^{x}) + 64 * {3}^{(2x)}({1}^{(8sin({3}^{x}))}((8cos({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})))ln(1) + \frac{(8sin({3}^{x}))(0)}{(1)}))ln^{2}(3)ln^{2}(1)cos^{2}({3}^{x}) + \frac{64 * {3}^{(2x)}{1}^{(8sin({3}^{x}))}*2ln(3)*0ln^{2}(1)cos^{2}({3}^{x})}{(3)} + \frac{64 * {3}^{(2x)}{1}^{(8sin({3}^{x}))}ln^{2}(3)*2ln(1)*0cos^{2}({3}^{x})}{(1)} + 64 * {3}^{(2x)}{1}^{(8sin({3}^{x}))}ln^{2}(3)ln^{2}(1)*-2cos({3}^{x})sin({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})) - 8({1}^{(8sin({3}^{x}))}((8cos({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})))ln(1) + \frac{(8sin({3}^{x}))(0)}{(1)})){3}^{(2x)}ln^{2}(3)ln(1)sin({3}^{x}) - 8 * {1}^{(8sin({3}^{x}))}({3}^{(2x)}((2)ln(3) + \frac{(2x)(0)}{(3)}))ln^{2}(3)ln(1)sin({3}^{x}) - \frac{8 * {1}^{(8sin({3}^{x}))}{3}^{(2x)}*2ln(3)*0ln(1)sin({3}^{x})}{(3)} - \frac{8 * {1}^{(8sin({3}^{x}))}{3}^{(2x)}ln^{2}(3)*0sin({3}^{x})}{(1)} - 8 * {1}^{(8sin({3}^{x}))}{3}^{(2x)}ln^{2}(3)ln(1)cos({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)}))\\=&8 * {3}^{x}{1}^{(8sin({3}^{x}))}ln^{3}(3)ln(1)cos({3}^{x}) + 192 * {3}^{(2x)}{1}^{(8sin({3}^{x}))}ln^{3}(3)ln^{2}(1)cos^{2}({3}^{x}) - 128 * {1}^{(8sin({3}^{x}))}{3}^{(3x)}ln^{3}(3)ln^{2}(1)sin({3}^{x})cos({3}^{x}) + 512 * {3}^{(3x)}{1}^{(8sin({3}^{x}))}ln^{3}(3)ln^{3}(1)cos^{3}({3}^{x}) - 64 * {3}^{(3x)}{1}^{(8sin({3}^{x}))}ln^{3}(3)ln^{2}(1)sin({3}^{x})cos({3}^{x}) - 8 * {1}^{(8sin({3}^{x}))}{3}^{(2x)}ln^{3}(3)ln(1)sin({3}^{x}) - 16 * {3}^{(2x)}{1}^{(8sin({3}^{x}))}ln^{3}(3)ln(1)sin({3}^{x}) - 8 * {1}^{(8sin({3}^{x}))}{3}^{(3x)}ln^{3}(3)ln(1)cos({3}^{x})\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( 8 * {3}^{x}{1}^{(8sin({3}^{x}))}ln^{3}(3)ln(1)cos({3}^{x}) + 192 * {3}^{(2x)}{1}^{(8sin({3}^{x}))}ln^{3}(3)ln^{2}(1)cos^{2}({3}^{x}) - 128 * {1}^{(8sin({3}^{x}))}{3}^{(3x)}ln^{3}(3)ln^{2}(1)sin({3}^{x})cos({3}^{x}) + 512 * {3}^{(3x)}{1}^{(8sin({3}^{x}))}ln^{3}(3)ln^{3}(1)cos^{3}({3}^{x}) - 64 * {3}^{(3x)}{1}^{(8sin({3}^{x}))}ln^{3}(3)ln^{2}(1)sin({3}^{x})cos({3}^{x}) - 8 * {1}^{(8sin({3}^{x}))}{3}^{(2x)}ln^{3}(3)ln(1)sin({3}^{x}) - 16 * {3}^{(2x)}{1}^{(8sin({3}^{x}))}ln^{3}(3)ln(1)sin({3}^{x}) - 8 * {1}^{(8sin({3}^{x}))}{3}^{(3x)}ln^{3}(3)ln(1)cos({3}^{x})\right)}{dx}\\=&8({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})){1}^{(8sin({3}^{x}))}ln^{3}(3)ln(1)cos({3}^{x}) + 8 * {3}^{x}({1}^{(8sin({3}^{x}))}((8cos({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})))ln(1) + \frac{(8sin({3}^{x}))(0)}{(1)}))ln^{3}(3)ln(1)cos({3}^{x}) + \frac{8 * {3}^{x}{1}^{(8sin({3}^{x}))}*3ln^{2}(3)*0ln(1)cos({3}^{x})}{(3)} + \frac{8 * {3}^{x}{1}^{(8sin({3}^{x}))}ln^{3}(3)*0cos({3}^{x})}{(1)} + 8 * {3}^{x}{1}^{(8sin({3}^{x}))}ln^{3}(3)ln(1)*-sin({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})) + 192({3}^{(2x)}((2)ln(3) + \frac{(2x)(0)}{(3)})){1}^{(8sin({3}^{x}))}ln^{3}(3)ln^{2}(1)cos^{2}({3}^{x}) + 192 * {3}^{(2x)}({1}^{(8sin({3}^{x}))}((8cos({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})))ln(1) + \frac{(8sin({3}^{x}))(0)}{(1)}))ln^{3}(3)ln^{2}(1)cos^{2}({3}^{x}) + \frac{192 * {3}^{(2x)}{1}^{(8sin({3}^{x}))}*3ln^{2}(3)*0ln^{2}(1)cos^{2}({3}^{x})}{(3)} + \frac{192 * {3}^{(2x)}{1}^{(8sin({3}^{x}))}ln^{3}(3)*2ln(1)*0cos^{2}({3}^{x})}{(1)} + 192 * {3}^{(2x)}{1}^{(8sin({3}^{x}))}ln^{3}(3)ln^{2}(1)*-2cos({3}^{x})sin({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})) - 128({1}^{(8sin({3}^{x}))}((8cos({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})))ln(1) + \frac{(8sin({3}^{x}))(0)}{(1)})){3}^{(3x)}ln^{3}(3)ln^{2}(1)sin({3}^{x})cos({3}^{x}) - 128 * {1}^{(8sin({3}^{x}))}({3}^{(3x)}((3)ln(3) + \frac{(3x)(0)}{(3)}))ln^{3}(3)ln^{2}(1)sin({3}^{x})cos({3}^{x}) - \frac{128 * {1}^{(8sin({3}^{x}))}{3}^{(3x)}*3ln^{2}(3)*0ln^{2}(1)sin({3}^{x})cos({3}^{x})}{(3)} - \frac{128 * {1}^{(8sin({3}^{x}))}{3}^{(3x)}ln^{3}(3)*2ln(1)*0sin({3}^{x})cos({3}^{x})}{(1)} - 128 * {1}^{(8sin({3}^{x}))}{3}^{(3x)}ln^{3}(3)ln^{2}(1)cos({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)}))cos({3}^{x}) - 128 * {1}^{(8sin({3}^{x}))}{3}^{(3x)}ln^{3}(3)ln^{2}(1)sin({3}^{x})*-sin({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})) + 512({3}^{(3x)}((3)ln(3) + \frac{(3x)(0)}{(3)})){1}^{(8sin({3}^{x}))}ln^{3}(3)ln^{3}(1)cos^{3}({3}^{x}) + 512 * {3}^{(3x)}({1}^{(8sin({3}^{x}))}((8cos({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})))ln(1) + \frac{(8sin({3}^{x}))(0)}{(1)}))ln^{3}(3)ln^{3}(1)cos^{3}({3}^{x}) + \frac{512 * {3}^{(3x)}{1}^{(8sin({3}^{x}))}*3ln^{2}(3)*0ln^{3}(1)cos^{3}({3}^{x})}{(3)} + \frac{512 * {3}^{(3x)}{1}^{(8sin({3}^{x}))}ln^{3}(3)*3ln^{2}(1)*0cos^{3}({3}^{x})}{(1)} + 512 * {3}^{(3x)}{1}^{(8sin({3}^{x}))}ln^{3}(3)ln^{3}(1)*-3cos^{2}({3}^{x})sin({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})) - 64({3}^{(3x)}((3)ln(3) + \frac{(3x)(0)}{(3)})){1}^{(8sin({3}^{x}))}ln^{3}(3)ln^{2}(1)sin({3}^{x})cos({3}^{x}) - 64 * {3}^{(3x)}({1}^{(8sin({3}^{x}))}((8cos({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})))ln(1) + \frac{(8sin({3}^{x}))(0)}{(1)}))ln^{3}(3)ln^{2}(1)sin({3}^{x})cos({3}^{x}) - \frac{64 * {3}^{(3x)}{1}^{(8sin({3}^{x}))}*3ln^{2}(3)*0ln^{2}(1)sin({3}^{x})cos({3}^{x})}{(3)} - \frac{64 * {3}^{(3x)}{1}^{(8sin({3}^{x}))}ln^{3}(3)*2ln(1)*0sin({3}^{x})cos({3}^{x})}{(1)} - 64 * {3}^{(3x)}{1}^{(8sin({3}^{x}))}ln^{3}(3)ln^{2}(1)cos({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)}))cos({3}^{x}) - 64 * {3}^{(3x)}{1}^{(8sin({3}^{x}))}ln^{3}(3)ln^{2}(1)sin({3}^{x})*-sin({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})) - 8({1}^{(8sin({3}^{x}))}((8cos({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})))ln(1) + \frac{(8sin({3}^{x}))(0)}{(1)})){3}^{(2x)}ln^{3}(3)ln(1)sin({3}^{x}) - 8 * {1}^{(8sin({3}^{x}))}({3}^{(2x)}((2)ln(3) + \frac{(2x)(0)}{(3)}))ln^{3}(3)ln(1)sin({3}^{x}) - \frac{8 * {1}^{(8sin({3}^{x}))}{3}^{(2x)}*3ln^{2}(3)*0ln(1)sin({3}^{x})}{(3)} - \frac{8 * {1}^{(8sin({3}^{x}))}{3}^{(2x)}ln^{3}(3)*0sin({3}^{x})}{(1)} - 8 * {1}^{(8sin({3}^{x}))}{3}^{(2x)}ln^{3}(3)ln(1)cos({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})) - 16({3}^{(2x)}((2)ln(3) + \frac{(2x)(0)}{(3)})){1}^{(8sin({3}^{x}))}ln^{3}(3)ln(1)sin({3}^{x}) - 16 * {3}^{(2x)}({1}^{(8sin({3}^{x}))}((8cos({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})))ln(1) + \frac{(8sin({3}^{x}))(0)}{(1)}))ln^{3}(3)ln(1)sin({3}^{x}) - \frac{16 * {3}^{(2x)}{1}^{(8sin({3}^{x}))}*3ln^{2}(3)*0ln(1)sin({3}^{x})}{(3)} - \frac{16 * {3}^{(2x)}{1}^{(8sin({3}^{x}))}ln^{3}(3)*0sin({3}^{x})}{(1)} - 16 * {3}^{(2x)}{1}^{(8sin({3}^{x}))}ln^{3}(3)ln(1)cos({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})) - 8({1}^{(8sin({3}^{x}))}((8cos({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)})))ln(1) + \frac{(8sin({3}^{x}))(0)}{(1)})){3}^{(3x)}ln^{3}(3)ln(1)cos({3}^{x}) - 8 * {1}^{(8sin({3}^{x}))}({3}^{(3x)}((3)ln(3) + \frac{(3x)(0)}{(3)}))ln^{3}(3)ln(1)cos({3}^{x}) - \frac{8 * {1}^{(8sin({3}^{x}))}{3}^{(3x)}*3ln^{2}(3)*0ln(1)cos({3}^{x})}{(3)} - \frac{8 * {1}^{(8sin({3}^{x}))}{3}^{(3x)}ln^{3}(3)*0cos({3}^{x})}{(1)} - 8 * {1}^{(8sin({3}^{x}))}{3}^{(3x)}ln^{3}(3)ln(1)*-sin({3}^{x})({3}^{x}((1)ln(3) + \frac{(x)(0)}{(3)}))\\=&8 * {3}^{x}{1}^{(8sin({3}^{x}))}ln^{4}(3)ln(1)cos({3}^{x}) + 448 * {3}^{(2x)}{1}^{(8sin({3}^{x}))}ln^{4}(3)ln^{2}(1)cos^{2}({3}^{x}) - 384 * {1}^{(8sin({3}^{x}))}{3}^{(3x)}ln^{4}(3)ln^{2}(1)sin({3}^{x})cos({3}^{x}) + 3072 * {3}^{(3x)}{1}^{(8sin({3}^{x}))}ln^{4}(3)ln^{3}(1)cos^{3}({3}^{x}) - 768 * {3}^{(3x)}{1}^{(8sin({3}^{x}))}ln^{4}(3)ln^{2}(1)sin({3}^{x})cos({3}^{x}) - 1536 * {3}^{(4x)}{1}^{(8sin({3}^{x}))}ln^{4}(3)ln^{3}(1)sin({3}^{x})cos^{2}({3}^{x}) - 1536 * {1}^{(8sin({3}^{x}))}{3}^{(4x)}ln^{4}(3)ln^{3}(1)sin({3}^{x})cos^{2}({3}^{x}) - 128 * {1}^{(8sin({3}^{x}))}{3}^{(4x)}ln^{4}(3)ln^{2}(1)cos^{2}({3}^{x}) + 128 * {3}^{(4x)}{1}^{(8sin({3}^{x}))}ln^{4}(3)ln^{2}(1)sin^{2}({3}^{x}) + 4096 * {3}^{(4x)}{1}^{(8sin({3}^{x}))}ln^{4}(3)ln^{4}(1)cos^{4}({3}^{x}) + 64 * {1}^{(8sin({3}^{x}))}{3}^{(4x)}ln^{4}(3)ln^{2}(1)sin^{2}({3}^{x}) - 128 * {3}^{(4x)}{1}^{(8sin({3}^{x}))}ln^{4}(3)ln^{2}(1)cos^{2}({3}^{x}) - 8 * {1}^{(8sin({3}^{x}))}{3}^{(2x)}ln^{4}(3)ln(1)sin({3}^{x}) - 48 * {3}^{(2x)}{1}^{(8sin({3}^{x}))}ln^{4}(3)ln(1)sin({3}^{x}) - 8 * {1}^{(8sin({3}^{x}))}{3}^{(3x)}ln^{4}(3)ln(1)cos({3}^{x}) - 40 * {3}^{(3x)}{1}^{(8sin({3}^{x}))}ln^{4}(3)ln(1)cos({3}^{x}) + 8 * {3}^{(4x)}{1}^{(8sin({3}^{x}))}ln^{4}(3)ln(1)sin({3}^{x})\\ \end{split}\end{equation} \]

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