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。\[ \begin{equation}\begin{split}【1/1】求函数\frac{sin(x)cos(x)}{sin(cos(x))} 关于 x 的 4 阶导数:\\\end{split}\end{equation} \]
\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{sin(x)cos(x)}{sin(cos(x))}\\&\color{blue}{函数的第 1 阶导数:}\\&\frac{d\left( \frac{sin(x)cos(x)}{sin(cos(x))}\right)}{dx}\\=&\frac{cos(x)cos(x)}{sin(cos(x))} + \frac{sin(x)*-cos(cos(x))*-sin(x)cos(x)}{sin^{2}(cos(x))} + \frac{sin(x)*-sin(x)}{sin(cos(x))}\\=&\frac{cos^{2}(x)}{sin(cos(x))} + \frac{sin^{2}(x)cos(cos(x))cos(x)}{sin^{2}(cos(x))} - \frac{sin^{2}(x)}{sin(cos(x))}\\\\ &\color{blue}{函数的第 2 阶导数:} \\&\frac{d\left( \frac{cos^{2}(x)}{sin(cos(x))} + \frac{sin^{2}(x)cos(cos(x))cos(x)}{sin^{2}(cos(x))} - \frac{sin^{2}(x)}{sin(cos(x))}\right)}{dx}\\=&\frac{-cos(cos(x))*-sin(x)cos^{2}(x)}{sin^{2}(cos(x))} + \frac{-2cos(x)sin(x)}{sin(cos(x))} + \frac{-2cos(cos(x))*-sin(x)sin^{2}(x)cos(cos(x))cos(x)}{sin^{3}(cos(x))} + \frac{2sin(x)cos(x)cos(cos(x))cos(x)}{sin^{2}(cos(x))} + \frac{sin^{2}(x)*-sin(cos(x))*-sin(x)cos(x)}{sin^{2}(cos(x))} + \frac{sin^{2}(x)cos(cos(x))*-sin(x)}{sin^{2}(cos(x))} - \frac{2sin(x)cos(x)}{sin(cos(x))} - \frac{sin^{2}(x)*-cos(cos(x))*-sin(x)}{sin^{2}(cos(x))}\\=&\frac{sin(x)cos(cos(x))cos^{2}(x)}{sin^{2}(cos(x))} + \frac{2sin(x)cos^{2}(x)cos(cos(x))}{sin^{2}(cos(x))} + \frac{2sin^{3}(x)cos^{2}(cos(x))cos(x)}{sin^{3}(cos(x))} - \frac{4sin(x)cos(x)}{sin(cos(x))} + \frac{sin^{3}(x)cos(x)}{sin(cos(x))} - \frac{2sin^{3}(x)cos(cos(x))}{sin^{2}(cos(x))}\\\\ &\color{blue}{函数的第 3 阶导数:} \\&\frac{d\left( \frac{sin(x)cos(cos(x))cos^{2}(x)}{sin^{2}(cos(x))} + \frac{2sin(x)cos^{2}(x)cos(cos(x))}{sin^{2}(cos(x))} + \frac{2sin^{3}(x)cos^{2}(cos(x))cos(x)}{sin^{3}(cos(x))} - \frac{4sin(x)cos(x)}{sin(cos(x))} + \frac{sin^{3}(x)cos(x)}{sin(cos(x))} - \frac{2sin^{3}(x)cos(cos(x))}{sin^{2}(cos(x))}\right)}{dx}\\=&\frac{cos(x)cos(cos(x))cos^{2}(x)}{sin^{2}(cos(x))} + \frac{sin(x)*-2cos(cos(x))*-sin(x)cos(cos(x))cos^{2}(x)}{sin^{3}(cos(x))} + \frac{sin(x)*-sin(cos(x))*-sin(x)cos^{2}(x)}{sin^{2}(cos(x))} + \frac{sin(x)cos(cos(x))*-2cos(x)sin(x)}{sin^{2}(cos(x))} + \frac{2cos(x)cos^{2}(x)cos(cos(x))}{sin^{2}(cos(x))} + \frac{2sin(x)*-2cos(cos(x))*-sin(x)cos^{2}(x)cos(cos(x))}{sin^{3}(cos(x))} + \frac{2sin(x)*-2cos(x)sin(x)cos(cos(x))}{sin^{2}(cos(x))} + \frac{2sin(x)cos^{2}(x)*-sin(cos(x))*-sin(x)}{sin^{2}(cos(x))} + \frac{2*3sin^{2}(x)cos(x)cos^{2}(cos(x))cos(x)}{sin^{3}(cos(x))} + \frac{2sin^{3}(x)*-3cos(cos(x))*-sin(x)cos^{2}(cos(x))cos(x)}{sin^{4}(cos(x))} + \frac{2sin^{3}(x)*-2cos(cos(x))sin(cos(x))*-sin(x)cos(x)}{sin^{3}(cos(x))} + \frac{2sin^{3}(x)cos^{2}(cos(x))*-sin(x)}{sin^{3}(cos(x))} - \frac{4cos(x)cos(x)}{sin(cos(x))} - \frac{4sin(x)*-cos(cos(x))*-sin(x)cos(x)}{sin^{2}(cos(x))} - \frac{4sin(x)*-sin(x)}{sin(cos(x))} + \frac{3sin^{2}(x)cos(x)cos(x)}{sin(cos(x))} + \frac{sin^{3}(x)*-cos(cos(x))*-sin(x)cos(x)}{sin^{2}(cos(x))} + \frac{sin^{3}(x)*-sin(x)}{sin(cos(x))} - \frac{2*-2cos(cos(x))*-sin(x)sin^{3}(x)cos(cos(x))}{sin^{3}(cos(x))} - \frac{2*3sin^{2}(x)cos(x)cos(cos(x))}{sin^{2}(cos(x))} - \frac{2sin^{3}(x)*-sin(cos(x))*-sin(x)}{sin^{2}(cos(x))}\\=&\frac{3cos^{3}(x)cos(cos(x))}{sin^{2}(cos(x))} + \frac{6sin^{2}(x)cos^{2}(cos(x))cos^{2}(x)}{sin^{3}(cos(x))} + \frac{5sin^{4}(x)cos(cos(x))cos(x)}{sin^{2}(cos(x))} - \frac{2sin^{2}(x)cos(cos(x))cos(x)}{sin^{2}(cos(x))} - \frac{10sin^{2}(x)cos(x)cos(cos(x))}{sin^{2}(cos(x))} + \frac{6sin^{2}(x)cos^{2}(x)cos^{2}(cos(x))}{sin^{3}(cos(x))} + \frac{6sin^{4}(x)cos^{3}(cos(x))cos(x)}{sin^{4}(cos(x))} - \frac{4sin^{2}(x)cos(cos(x))cos(x)}{sin^{2}(cos(x))} - \frac{6sin^{4}(x)cos^{2}(cos(x))}{sin^{3}(cos(x))} - \frac{4cos^{2}(x)}{sin(cos(x))} + \frac{3sin^{2}(x)cos^{2}(x)}{sin(cos(x))} + \frac{3sin^{2}(x)cos^{2}(x)}{sin(cos(x))} + \frac{4sin^{2}(x)}{sin(cos(x))} - \frac{3sin^{4}(x)}{sin(cos(x))}\\\\ &\color{blue}{函数的第 4 阶导数:} \\&\frac{d\left( \frac{3cos^{3}(x)cos(cos(x))}{sin^{2}(cos(x))} + \frac{6sin^{2}(x)cos^{2}(cos(x))cos^{2}(x)}{sin^{3}(cos(x))} + \frac{5sin^{4}(x)cos(cos(x))cos(x)}{sin^{2}(cos(x))} - \frac{2sin^{2}(x)cos(cos(x))cos(x)}{sin^{2}(cos(x))} - \frac{10sin^{2}(x)cos(x)cos(cos(x))}{sin^{2}(cos(x))} + \frac{6sin^{2}(x)cos^{2}(x)cos^{2}(cos(x))}{sin^{3}(cos(x))} + \frac{6sin^{4}(x)cos^{3}(cos(x))cos(x)}{sin^{4}(cos(x))} - \frac{4sin^{2}(x)cos(cos(x))cos(x)}{sin^{2}(cos(x))} - \frac{6sin^{4}(x)cos^{2}(cos(x))}{sin^{3}(cos(x))} - \frac{4cos^{2}(x)}{sin(cos(x))} + \frac{3sin^{2}(x)cos^{2}(x)}{sin(cos(x))} + \frac{3sin^{2}(x)cos^{2}(x)}{sin(cos(x))} + \frac{4sin^{2}(x)}{sin(cos(x))} - \frac{3sin^{4}(x)}{sin(cos(x))}\right)}{dx}\\=&\frac{3*-2cos(cos(x))*-sin(x)cos^{3}(x)cos(cos(x))}{sin^{3}(cos(x))} + \frac{3*-3cos^{2}(x)sin(x)cos(cos(x))}{sin^{2}(cos(x))} + \frac{3cos^{3}(x)*-sin(cos(x))*-sin(x)}{sin^{2}(cos(x))} + \frac{6*-3cos(cos(x))*-sin(x)sin^{2}(x)cos^{2}(cos(x))cos^{2}(x)}{sin^{4}(cos(x))} + \frac{6*2sin(x)cos(x)cos^{2}(cos(x))cos^{2}(x)}{sin^{3}(cos(x))} + \frac{6sin^{2}(x)*-2cos(cos(x))sin(cos(x))*-sin(x)cos^{2}(x)}{sin^{3}(cos(x))} + \frac{6sin^{2}(x)cos^{2}(cos(x))*-2cos(x)sin(x)}{sin^{3}(cos(x))} + \frac{5*-2cos(cos(x))*-sin(x)sin^{4}(x)cos(cos(x))cos(x)}{sin^{3}(cos(x))} + \frac{5*4sin^{3}(x)cos(x)cos(cos(x))cos(x)}{sin^{2}(cos(x))} + \frac{5sin^{4}(x)*-sin(cos(x))*-sin(x)cos(x)}{sin^{2}(cos(x))} + \frac{5sin^{4}(x)cos(cos(x))*-sin(x)}{sin^{2}(cos(x))} - \frac{2*2sin(x)cos(x)cos(cos(x))cos(x)}{sin^{2}(cos(x))} - \frac{2sin^{2}(x)*-2cos(cos(x))*-sin(x)cos(cos(x))cos(x)}{sin^{3}(cos(x))} - \frac{2sin^{2}(x)*-sin(cos(x))*-sin(x)cos(x)}{sin^{2}(cos(x))} - \frac{2sin^{2}(x)cos(cos(x))*-sin(x)}{sin^{2}(cos(x))} - \frac{10*2sin(x)cos(x)cos(x)cos(cos(x))}{sin^{2}(cos(x))} - \frac{10sin^{2}(x)*-2cos(cos(x))*-sin(x)cos(x)cos(cos(x))}{sin^{3}(cos(x))} - \frac{10sin^{2}(x)*-sin(x)cos(cos(x))}{sin^{2}(cos(x))} - \frac{10sin^{2}(x)cos(x)*-sin(cos(x))*-sin(x)}{sin^{2}(cos(x))} + \frac{6*2sin(x)cos(x)cos^{2}(x)cos^{2}(cos(x))}{sin^{3}(cos(x))} + \frac{6sin^{2}(x)*-3cos(cos(x))*-sin(x)cos^{2}(x)cos^{2}(cos(x))}{sin^{4}(cos(x))} + \frac{6sin^{2}(x)*-2cos(x)sin(x)cos^{2}(cos(x))}{sin^{3}(cos(x))} + \frac{6sin^{2}(x)cos^{2}(x)*-2cos(cos(x))sin(cos(x))*-sin(x)}{sin^{3}(cos(x))} + \frac{6*-4cos(cos(x))*-sin(x)sin^{4}(x)cos^{3}(cos(x))cos(x)}{sin^{5}(cos(x))} + \frac{6*4sin^{3}(x)cos(x)cos^{3}(cos(x))cos(x)}{sin^{4}(cos(x))} + \frac{6sin^{4}(x)*-3cos^{2}(cos(x))sin(cos(x))*-sin(x)cos(x)}{sin^{4}(cos(x))} + \frac{6sin^{4}(x)cos^{3}(cos(x))*-sin(x)}{sin^{4}(cos(x))} - \frac{4*-2cos(cos(x))*-sin(x)sin^{2}(x)cos(cos(x))cos(x)}{sin^{3}(cos(x))} - \frac{4*2sin(x)cos(x)cos(cos(x))cos(x)}{sin^{2}(cos(x))} - \frac{4sin^{2}(x)*-sin(cos(x))*-sin(x)cos(x)}{sin^{2}(cos(x))} - \frac{4sin^{2}(x)cos(cos(x))*-sin(x)}{sin^{2}(cos(x))} - \frac{6*4sin^{3}(x)cos(x)cos^{2}(cos(x))}{sin^{3}(cos(x))} - \frac{6sin^{4}(x)*-3cos(cos(x))*-sin(x)cos^{2}(cos(x))}{sin^{4}(cos(x))} - \frac{6sin^{4}(x)*-2cos(cos(x))sin(cos(x))*-sin(x)}{sin^{3}(cos(x))} - \frac{4*-cos(cos(x))*-sin(x)cos^{2}(x)}{sin^{2}(cos(x))} - \frac{4*-2cos(x)sin(x)}{sin(cos(x))} + \frac{3*-cos(cos(x))*-sin(x)sin^{2}(x)cos^{2}(x)}{sin^{2}(cos(x))} + \frac{3*2sin(x)cos(x)cos^{2}(x)}{sin(cos(x))} + \frac{3sin^{2}(x)*-2cos(x)sin(x)}{sin(cos(x))} + \frac{3*2sin(x)cos(x)cos^{2}(x)}{sin(cos(x))} + \frac{3sin^{2}(x)*-cos(cos(x))*-sin(x)cos^{2}(x)}{sin^{2}(cos(x))} + \frac{3sin^{2}(x)*-2cos(x)sin(x)}{sin(cos(x))} + \frac{4*2sin(x)cos(x)}{sin(cos(x))} + \frac{4sin^{2}(x)*-cos(cos(x))*-sin(x)}{sin^{2}(cos(x))} - \frac{3*4sin^{3}(x)cos(x)}{sin(cos(x))} - \frac{3sin^{4}(x)*-cos(cos(x))*-sin(x)}{sin^{2}(cos(x))}\\=&\frac{6sin(x)cos^{2}(cos(x))cos^{3}(x)}{sin^{3}(cos(x))} + \frac{24sin(x)cos^{3}(x)cos^{2}(cos(x))}{sin^{3}(cos(x))} + \frac{18sin^{3}(x)cos^{3}(cos(x))cos^{2}(x)}{sin^{4}(cos(x))} + \frac{28sin^{5}(x)cos^{2}(cos(x))cos(x)}{sin^{3}(cos(x))} + \frac{20sin^{3}(x)cos^{2}(x)cos(cos(x))}{sin^{2}(cos(x))} + \frac{15sin^{3}(x)cos(cos(x))cos^{2}(x)}{sin^{2}(cos(x))} - \frac{36sin^{3}(x)cos^{2}(cos(x))cos(x)}{sin^{3}(cos(x))} - \frac{41sin(x)cos^{2}(x)cos(cos(x))}{sin^{2}(cos(x))} + \frac{15sin^{3}(x)cos(cos(x))cos^{2}(x)}{sin^{2}(cos(x))} - \frac{36sin^{3}(x)cos(x)cos^{2}(cos(x))}{sin^{3}(cos(x))} + \frac{18sin^{3}(x)cos^{3}(cos(x))cos^{2}(x)}{sin^{4}(cos(x))} + \frac{24sin^{3}(x)cos^{2}(x)cos^{3}(cos(x))}{sin^{4}(cos(x))} + \frac{24sin^{5}(x)cos^{4}(cos(x))cos(x)}{sin^{5}(cos(x))} - \frac{8sin^{3}(x)cos^{2}(cos(x))cos(x)}{sin^{3}(cos(x))} - \frac{4sin(x)cos(cos(x))cos^{2}(x)}{sin^{2}(cos(x))} - \frac{20sin^{5}(x)cos(cos(x))}{sin^{2}(cos(x))} + \frac{5sin^{5}(x)cos(x)}{sin(cos(x))} + \frac{12sin^{3}(x)cos(cos(x))}{sin^{2}(cos(x))} - \frac{24sin^{5}(x)cos^{3}(cos(x))}{sin^{4}(cos(x))} - \frac{22sin^{3}(x)cos(x)}{sin(cos(x))} + \frac{8sin^{3}(x)cos(cos(x))}{sin^{2}(cos(x))} - \frac{18sin^{3}(x)cos(x)}{sin(cos(x))} + \frac{12sin(x)cos^{3}(x)}{sin(cos(x))} + \frac{16sin(x)cos(x)}{sin(cos(x))} + \frac{3sin(x)cos^{3}(x)}{sin(cos(x))}\\ \end{split}\end{equation} \]你的问题在这里没有得到解决?请到 热门难题 里面看看吧!