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。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{({(sin(x))}^{7} + {(cos(x))}^{7})}{tan(x)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{sin^{7}(x)}{tan(x)} + \frac{cos^{7}(x)}{tan(x)}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{sin^{7}(x)}{tan(x)} + \frac{cos^{7}(x)}{tan(x)}\right)}{dx}\\=&\frac{7sin^{6}(x)cos(x)}{tan(x)} + \frac{sin^{7}(x)*-sec^{2}(x)(1)}{tan^{2}(x)} + \frac{-7cos^{6}(x)sin(x)}{tan(x)} + \frac{cos^{7}(x)*-sec^{2}(x)(1)}{tan^{2}(x)}\\=&\frac{7sin^{6}(x)cos(x)}{tan(x)} - \frac{sin^{7}(x)sec^{2}(x)}{tan^{2}(x)} - \frac{7sin(x)cos^{6}(x)}{tan(x)} - \frac{cos^{7}(x)sec^{2}(x)}{tan^{2}(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{7sin^{6}(x)cos(x)}{tan(x)} - \frac{sin^{7}(x)sec^{2}(x)}{tan^{2}(x)} - \frac{7sin(x)cos^{6}(x)}{tan(x)} - \frac{cos^{7}(x)sec^{2}(x)}{tan^{2}(x)}\right)}{dx}\\=&\frac{7*6sin^{5}(x)cos(x)cos(x)}{tan(x)} + \frac{7sin^{6}(x)*-sin(x)}{tan(x)} + \frac{7sin^{6}(x)cos(x)*-sec^{2}(x)(1)}{tan^{2}(x)} - \frac{7sin^{6}(x)cos(x)sec^{2}(x)}{tan^{2}(x)} - \frac{sin^{7}(x)*-2sec^{2}(x)(1)sec^{2}(x)}{tan^{3}(x)} - \frac{sin^{7}(x)*2sec^{2}(x)tan(x)}{tan^{2}(x)} - \frac{7cos(x)cos^{6}(x)}{tan(x)} - \frac{7sin(x)*-6cos^{5}(x)sin(x)}{tan(x)} - \frac{7sin(x)cos^{6}(x)*-sec^{2}(x)(1)}{tan^{2}(x)} - \frac{-7cos^{6}(x)sin(x)sec^{2}(x)}{tan^{2}(x)} - \frac{cos^{7}(x)*-2sec^{2}(x)(1)sec^{2}(x)}{tan^{3}(x)} - \frac{cos^{7}(x)*2sec^{2}(x)tan(x)}{tan^{2}(x)}\\=&\frac{-14sin^{6}(x)cos(x)sec^{2}(x)}{tan^{2}(x)} + \frac{2sin^{7}(x)sec^{4}(x)}{tan^{3}(x)} + \frac{14sin(x)cos^{6}(x)sec^{2}(x)}{tan^{2}(x)} - \frac{2sin^{7}(x)sec^{2}(x)}{tan(x)} - \frac{7sin^{7}(x)}{tan(x)} + \frac{2cos^{7}(x)sec^{4}(x)}{tan^{3}(x)} + \frac{42sin^{2}(x)cos^{5}(x)}{tan(x)} + \frac{42sin^{5}(x)cos^{2}(x)}{tan(x)} - \frac{2cos^{7}(x)sec^{2}(x)}{tan(x)} - \frac{7cos^{7}(x)}{tan(x)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-14sin^{6}(x)cos(x)sec^{2}(x)}{tan^{2}(x)} + \frac{2sin^{7}(x)sec^{4}(x)}{tan^{3}(x)} + \frac{14sin(x)cos^{6}(x)sec^{2}(x)}{tan^{2}(x)} - \frac{2sin^{7}(x)sec^{2}(x)}{tan(x)} - \frac{7sin^{7}(x)}{tan(x)} + \frac{2cos^{7}(x)sec^{4}(x)}{tan^{3}(x)} + \frac{42sin^{2}(x)cos^{5}(x)}{tan(x)} + \frac{42sin^{5}(x)cos^{2}(x)}{tan(x)} - \frac{2cos^{7}(x)sec^{2}(x)}{tan(x)} - \frac{7cos^{7}(x)}{tan(x)}\right)}{dx}\\=&\frac{-14*6sin^{5}(x)cos(x)cos(x)sec^{2}(x)}{tan^{2}(x)} - \frac{14sin^{6}(x)*-sin(x)sec^{2}(x)}{tan^{2}(x)} - \frac{14sin^{6}(x)cos(x)*-2sec^{2}(x)(1)sec^{2}(x)}{tan^{3}(x)} - \frac{14sin^{6}(x)cos(x)*2sec^{2}(x)tan(x)}{tan^{2}(x)} + \frac{2*7sin^{6}(x)cos(x)sec^{4}(x)}{tan^{3}(x)} + \frac{2sin^{7}(x)*-3sec^{2}(x)(1)sec^{4}(x)}{tan^{4}(x)} + \frac{2sin^{7}(x)*4sec^{4}(x)tan(x)}{tan^{3}(x)} + \frac{14cos(x)cos^{6}(x)sec^{2}(x)}{tan^{2}(x)} + \frac{14sin(x)*-6cos^{5}(x)sin(x)sec^{2}(x)}{tan^{2}(x)} + \frac{14sin(x)cos^{6}(x)*-2sec^{2}(x)(1)sec^{2}(x)}{tan^{3}(x)} + \frac{14sin(x)cos^{6}(x)*2sec^{2}(x)tan(x)}{tan^{2}(x)} - \frac{2*7sin^{6}(x)cos(x)sec^{2}(x)}{tan(x)} - \frac{2sin^{7}(x)*-sec^{2}(x)(1)sec^{2}(x)}{tan^{2}(x)} - \frac{2sin^{7}(x)*2sec^{2}(x)tan(x)}{tan(x)} - \frac{7*7sin^{6}(x)cos(x)}{tan(x)} - \frac{7sin^{7}(x)*-sec^{2}(x)(1)}{tan^{2}(x)} + \frac{2*-7cos^{6}(x)sin(x)sec^{4}(x)}{tan^{3}(x)} + \frac{2cos^{7}(x)*-3sec^{2}(x)(1)sec^{4}(x)}{tan^{4}(x)} + \frac{2cos^{7}(x)*4sec^{4}(x)tan(x)}{tan^{3}(x)} + \frac{42*2sin(x)cos(x)cos^{5}(x)}{tan(x)} + \frac{42sin^{2}(x)*-5cos^{4}(x)sin(x)}{tan(x)} + \frac{42sin^{2}(x)cos^{5}(x)*-sec^{2}(x)(1)}{tan^{2}(x)} + \frac{42*5sin^{4}(x)cos(x)cos^{2}(x)}{tan(x)} + \frac{42sin^{5}(x)*-2cos(x)sin(x)}{tan(x)} + \frac{42sin^{5}(x)cos^{2}(x)*-sec^{2}(x)(1)}{tan^{2}(x)} - \frac{2*-7cos^{6}(x)sin(x)sec^{2}(x)}{tan(x)} - \frac{2cos^{7}(x)*-sec^{2}(x)(1)sec^{2}(x)}{tan^{2}(x)} - \frac{2cos^{7}(x)*2sec^{2}(x)tan(x)}{tan(x)} - \frac{7*-7cos^{6}(x)sin(x)}{tan(x)} - \frac{7cos^{7}(x)*-sec^{2}(x)(1)}{tan^{2}(x)}\\=&\frac{-126sin^{5}(x)cos^{2}(x)sec^{2}(x)}{tan^{2}(x)} + \frac{21sin^{7}(x)sec^{2}(x)}{tan^{2}(x)} + \frac{42sin^{6}(x)cos(x)sec^{4}(x)}{tan^{3}(x)} - \frac{42sin^{6}(x)cos(x)sec^{2}(x)}{tan(x)} - \frac{42sin(x)cos^{6}(x)sec^{4}(x)}{tan^{3}(x)} - \frac{6sin^{7}(x)sec^{6}(x)}{tan^{4}(x)} + \frac{10sin^{7}(x)sec^{4}(x)}{tan^{2}(x)} + \frac{21cos^{7}(x)sec^{2}(x)}{tan^{2}(x)} - \frac{126sin^{2}(x)cos^{5}(x)sec^{2}(x)}{tan^{2}(x)} + \frac{42sin(x)cos^{6}(x)sec^{2}(x)}{tan(x)} - 4sin^{7}(x)sec^{2}(x) - \frac{6cos^{7}(x)sec^{6}(x)}{tan^{4}(x)} + \frac{10cos^{7}(x)sec^{4}(x)}{tan^{2}(x)} + \frac{133sin(x)cos^{6}(x)}{tan(x)} - \frac{210sin^{3}(x)cos^{4}(x)}{tan(x)} + \frac{210sin^{4}(x)cos^{3}(x)}{tan(x)} - \frac{133sin^{6}(x)cos(x)}{tan(x)} - 4cos^{7}(x)sec^{2}(x)\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-126sin^{5}(x)cos^{2}(x)sec^{2}(x)}{tan^{2}(x)} + \frac{21sin^{7}(x)sec^{2}(x)}{tan^{2}(x)} + \frac{42sin^{6}(x)cos(x)sec^{4}(x)}{tan^{3}(x)} - \frac{42sin^{6}(x)cos(x)sec^{2}(x)}{tan(x)} - \frac{42sin(x)cos^{6}(x)sec^{4}(x)}{tan^{3}(x)} - \frac{6sin^{7}(x)sec^{6}(x)}{tan^{4}(x)} + \frac{10sin^{7}(x)sec^{4}(x)}{tan^{2}(x)} + \frac{21cos^{7}(x)sec^{2}(x)}{tan^{2}(x)} - \frac{126sin^{2}(x)cos^{5}(x)sec^{2}(x)}{tan^{2}(x)} + \frac{42sin(x)cos^{6}(x)sec^{2}(x)}{tan(x)} - 4sin^{7}(x)sec^{2}(x) - \frac{6cos^{7}(x)sec^{6}(x)}{tan^{4}(x)} + \frac{10cos^{7}(x)sec^{4}(x)}{tan^{2}(x)} + \frac{133sin(x)cos^{6}(x)}{tan(x)} - \frac{210sin^{3}(x)cos^{4}(x)}{tan(x)} + \frac{210sin^{4}(x)cos^{3}(x)}{tan(x)} - \frac{133sin^{6}(x)cos(x)}{tan(x)} - 4cos^{7}(x)sec^{2}(x)\right)}{dx}\\=&\frac{-126*5sin^{4}(x)cos(x)cos^{2}(x)sec^{2}(x)}{tan^{2}(x)} - \frac{126sin^{5}(x)*-2cos(x)sin(x)sec^{2}(x)}{tan^{2}(x)} - \frac{126sin^{5}(x)cos^{2}(x)*-2sec^{2}(x)(1)sec^{2}(x)}{tan^{3}(x)} - \frac{126sin^{5}(x)cos^{2}(x)*2sec^{2}(x)tan(x)}{tan^{2}(x)} + \frac{21*7sin^{6}(x)cos(x)sec^{2}(x)}{tan^{2}(x)} + \frac{21sin^{7}(x)*-2sec^{2}(x)(1)sec^{2}(x)}{tan^{3}(x)} + \frac{21sin^{7}(x)*2sec^{2}(x)tan(x)}{tan^{2}(x)} + \frac{42*6sin^{5}(x)cos(x)cos(x)sec^{4}(x)}{tan^{3}(x)} + \frac{42sin^{6}(x)*-sin(x)sec^{4}(x)}{tan^{3}(x)} + \frac{42sin^{6}(x)cos(x)*-3sec^{2}(x)(1)sec^{4}(x)}{tan^{4}(x)} + \frac{42sin^{6}(x)cos(x)*4sec^{4}(x)tan(x)}{tan^{3}(x)} - \frac{42*6sin^{5}(x)cos(x)cos(x)sec^{2}(x)}{tan(x)} - \frac{42sin^{6}(x)*-sin(x)sec^{2}(x)}{tan(x)} - \frac{42sin^{6}(x)cos(x)*-sec^{2}(x)(1)sec^{2}(x)}{tan^{2}(x)} - \frac{42sin^{6}(x)cos(x)*2sec^{2}(x)tan(x)}{tan(x)} - \frac{42cos(x)cos^{6}(x)sec^{4}(x)}{tan^{3}(x)} - \frac{42sin(x)*-6cos^{5}(x)sin(x)sec^{4}(x)}{tan^{3}(x)} - \frac{42sin(x)cos^{6}(x)*-3sec^{2}(x)(1)sec^{4}(x)}{tan^{4}(x)} - \frac{42sin(x)cos^{6}(x)*4sec^{4}(x)tan(x)}{tan^{3}(x)} - \frac{6*7sin^{6}(x)cos(x)sec^{6}(x)}{tan^{4}(x)} - \frac{6sin^{7}(x)*-4sec^{2}(x)(1)sec^{6}(x)}{tan^{5}(x)} - \frac{6sin^{7}(x)*6sec^{6}(x)tan(x)}{tan^{4}(x)} + \frac{10*7sin^{6}(x)cos(x)sec^{4}(x)}{tan^{2}(x)} + \frac{10sin^{7}(x)*-2sec^{2}(x)(1)sec^{4}(x)}{tan^{3}(x)} + \frac{10sin^{7}(x)*4sec^{4}(x)tan(x)}{tan^{2}(x)} + \frac{21*-7cos^{6}(x)sin(x)sec^{2}(x)}{tan^{2}(x)} + \frac{21cos^{7}(x)*-2sec^{2}(x)(1)sec^{2}(x)}{tan^{3}(x)} + \frac{21cos^{7}(x)*2sec^{2}(x)tan(x)}{tan^{2}(x)} - \frac{126*2sin(x)cos(x)cos^{5}(x)sec^{2}(x)}{tan^{2}(x)} - \frac{126sin^{2}(x)*-5cos^{4}(x)sin(x)sec^{2}(x)}{tan^{2}(x)} - \frac{126sin^{2}(x)cos^{5}(x)*-2sec^{2}(x)(1)sec^{2}(x)}{tan^{3}(x)} - \frac{126sin^{2}(x)cos^{5}(x)*2sec^{2}(x)tan(x)}{tan^{2}(x)} + \frac{42cos(x)cos^{6}(x)sec^{2}(x)}{tan(x)} + \frac{42sin(x)*-6cos^{5}(x)sin(x)sec^{2}(x)}{tan(x)} + \frac{42sin(x)cos^{6}(x)*-sec^{2}(x)(1)sec^{2}(x)}{tan^{2}(x)} + \frac{42sin(x)cos^{6}(x)*2sec^{2}(x)tan(x)}{tan(x)} - 4*7sin^{6}(x)cos(x)sec^{2}(x) - 4sin^{7}(x)*2sec^{2}(x)tan(x) - \frac{6*-7cos^{6}(x)sin(x)sec^{6}(x)}{tan^{4}(x)} - \frac{6cos^{7}(x)*-4sec^{2}(x)(1)sec^{6}(x)}{tan^{5}(x)} - \frac{6cos^{7}(x)*6sec^{6}(x)tan(x)}{tan^{4}(x)} + \frac{10*-7cos^{6}(x)sin(x)sec^{4}(x)}{tan^{2}(x)} + \frac{10cos^{7}(x)*-2sec^{2}(x)(1)sec^{4}(x)}{tan^{3}(x)} + \frac{10cos^{7}(x)*4sec^{4}(x)tan(x)}{tan^{2}(x)} + \frac{133cos(x)cos^{6}(x)}{tan(x)} + \frac{133sin(x)*-6cos^{5}(x)sin(x)}{tan(x)} + \frac{133sin(x)cos^{6}(x)*-sec^{2}(x)(1)}{tan^{2}(x)} - \frac{210*3sin^{2}(x)cos(x)cos^{4}(x)}{tan(x)} - \frac{210sin^{3}(x)*-4cos^{3}(x)sin(x)}{tan(x)} - \frac{210sin^{3}(x)cos^{4}(x)*-sec^{2}(x)(1)}{tan^{2}(x)} + \frac{210*4sin^{3}(x)cos(x)cos^{3}(x)}{tan(x)} + \frac{210sin^{4}(x)*-3cos^{2}(x)sin(x)}{tan(x)} + \frac{210sin^{4}(x)cos^{3}(x)*-sec^{2}(x)(1)}{tan^{2}(x)} - \frac{133*6sin^{5}(x)cos(x)cos(x)}{tan(x)} - \frac{133sin^{6}(x)*-sin(x)}{tan(x)} - \frac{133sin^{6}(x)cos(x)*-sec^{2}(x)(1)}{tan^{2}(x)} - 4*-7cos^{6}(x)sin(x)sec^{2}(x) - 4cos^{7}(x)*2sec^{2}(x)tan(x)\\=&\frac{-840sin^{4}(x)cos^{3}(x)sec^{2}(x)}{tan^{2}(x)} + \frac{532sin^{6}(x)cos(x)sec^{2}(x)}{tan^{2}(x)} + \frac{504sin^{5}(x)cos^{2}(x)sec^{4}(x)}{tan^{3}(x)} - \frac{504sin^{5}(x)cos^{2}(x)sec^{2}(x)}{tan(x)} - \frac{168sin^{6}(x)cos(x)sec^{6}(x)}{tan^{4}(x)} - \frac{84sin^{7}(x)sec^{4}(x)}{tan^{3}(x)} + \frac{84sin^{7}(x)sec^{2}(x)}{tan(x)} + \frac{280sin^{6}(x)cos(x)sec^{4}(x)}{tan^{2}(x)} + \frac{168sin(x)cos^{6}(x)sec^{6}(x)}{tan^{4}(x)} - 112sin^{6}(x)cos(x)sec^{2}(x) - \frac{84cos^{7}(x)sec^{4}(x)}{tan^{3}(x)} + \frac{504sin^{2}(x)cos^{5}(x)sec^{4}(x)}{tan^{3}(x)} - \frac{280sin(x)cos^{6}(x)sec^{4}(x)}{tan^{2}(x)} + \frac{24sin^{7}(x)sec^{8}(x)}{tan^{5}(x)} - \frac{56sin^{7}(x)sec^{6}(x)}{tan^{3}(x)} + \frac{40sin^{7}(x)sec^{4}(x)}{tan(x)} - \frac{532sin(x)cos^{6}(x)sec^{2}(x)}{tan^{2}(x)} + \frac{84cos^{7}(x)sec^{2}(x)}{tan(x)} + \frac{840sin^{3}(x)cos^{4}(x)sec^{2}(x)}{tan^{2}(x)} - \frac{504sin^{2}(x)cos^{5}(x)sec^{2}(x)}{tan(x)} + 112sin(x)cos^{6}(x)sec^{2}(x) - 8sin^{7}(x)tan(x)sec^{2}(x) + \frac{24cos^{7}(x)sec^{8}(x)}{tan^{5}(x)} - \frac{56cos^{7}(x)sec^{6}(x)}{tan^{3}(x)} + \frac{40cos^{7}(x)sec^{4}(x)}{tan(x)} - 8cos^{7}(x)tan(x)sec^{2}(x) - \frac{1428sin^{2}(x)cos^{5}(x)}{tan(x)} + \frac{840sin^{4}(x)cos^{3}(x)}{tan(x)} + \frac{840sin^{3}(x)cos^{4}(x)}{tan(x)} - \frac{1428sin^{5}(x)cos^{2}(x)}{tan(x)} + \frac{133sin^{7}(x)}{tan(x)} + \frac{133cos^{7}(x)}{tan(x)}\\ \end{split}\end{equation} \]

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