There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ 6x + \frac{sin(x)cos(x)}{tan(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( 6x + \frac{sin(x)cos(x)}{tan(x)}\right)}{dx}\\=&6 + \frac{cos(x)cos(x)}{tan(x)} + \frac{sin(x)*-sin(x)}{tan(x)} + \frac{sin(x)cos(x)*-sec^{2}(x)(1)}{tan^{2}(x)}\\=&\frac{cos^{2}(x)}{tan(x)} - \frac{sin^{2}(x)}{tan(x)} - \frac{sin(x)cos(x)sec^{2}(x)}{tan^{2}(x)} + 6\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{cos^{2}(x)}{tan(x)} - \frac{sin^{2}(x)}{tan(x)} - \frac{sin(x)cos(x)sec^{2}(x)}{tan^{2}(x)} + 6\right)}{dx}\\=&\frac{-2cos(x)sin(x)}{tan(x)} + \frac{cos^{2}(x)*-sec^{2}(x)(1)}{tan^{2}(x)} - \frac{2sin(x)cos(x)}{tan(x)} - \frac{sin^{2}(x)*-sec^{2}(x)(1)}{tan^{2}(x)} - \frac{cos(x)cos(x)sec^{2}(x)}{tan^{2}(x)} - \frac{sin(x)*-sin(x)sec^{2}(x)}{tan^{2}(x)} - \frac{sin(x)cos(x)*-2sec^{2}(x)(1)sec^{2}(x)}{tan^{3}(x)} - \frac{sin(x)cos(x)*2sec^{2}(x)tan(x)}{tan^{2}(x)} + 0\\=&\frac{2sin(x)cos(x)sec^{4}(x)}{tan^{3}(x)} - \frac{2cos^{2}(x)sec^{2}(x)}{tan^{2}(x)} - \frac{2sin(x)cos(x)sec^{2}(x)}{tan(x)} + \frac{2sin^{2}(x)sec^{2}(x)}{tan^{2}(x)} - \frac{4sin(x)cos(x)}{tan(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2sin(x)cos(x)sec^{4}(x)}{tan^{3}(x)} - \frac{2cos^{2}(x)sec^{2}(x)}{tan^{2}(x)} - \frac{2sin(x)cos(x)sec^{2}(x)}{tan(x)} + \frac{2sin^{2}(x)sec^{2}(x)}{tan^{2}(x)} - \frac{4sin(x)cos(x)}{tan(x)}\right)}{dx}\\=&\frac{2cos(x)cos(x)sec^{4}(x)}{tan^{3}(x)} + \frac{2sin(x)*-sin(x)sec^{4}(x)}{tan^{3}(x)} + \frac{2sin(x)cos(x)*-3sec^{2}(x)(1)sec^{4}(x)}{tan^{4}(x)} + \frac{2sin(x)cos(x)*4sec^{4}(x)tan(x)}{tan^{3}(x)} - \frac{2*-2cos(x)sin(x)sec^{2}(x)}{tan^{2}(x)} - \frac{2cos^{2}(x)*-2sec^{2}(x)(1)sec^{2}(x)}{tan^{3}(x)} - \frac{2cos^{2}(x)*2sec^{2}(x)tan(x)}{tan^{2}(x)} - \frac{2cos(x)cos(x)sec^{2}(x)}{tan(x)} - \frac{2sin(x)*-sin(x)sec^{2}(x)}{tan(x)} - \frac{2sin(x)cos(x)*-sec^{2}(x)(1)sec^{2}(x)}{tan^{2}(x)} - \frac{2sin(x)cos(x)*2sec^{2}(x)tan(x)}{tan(x)} + \frac{2*2sin(x)cos(x)sec^{2}(x)}{tan^{2}(x)} + \frac{2sin^{2}(x)*-2sec^{2}(x)(1)sec^{2}(x)}{tan^{3}(x)} + \frac{2sin^{2}(x)*2sec^{2}(x)tan(x)}{tan^{2}(x)} - \frac{4cos(x)cos(x)}{tan(x)} - \frac{4sin(x)*-sin(x)}{tan(x)} - \frac{4sin(x)cos(x)*-sec^{2}(x)(1)}{tan^{2}(x)}\\=&\frac{6cos^{2}(x)sec^{4}(x)}{tan^{3}(x)} - \frac{6sin^{2}(x)sec^{4}(x)}{tan^{3}(x)} - \frac{6sin(x)cos(x)sec^{6}(x)}{tan^{4}(x)} + \frac{10sin(x)cos(x)sec^{4}(x)}{tan^{2}(x)} - \frac{6cos^{2}(x)sec^{2}(x)}{tan(x)} + \frac{6sin^{2}(x)sec^{2}(x)}{tan(x)} + \frac{12sin(x)cos(x)sec^{2}(x)}{tan^{2}(x)} - 4sin(x)cos(x)sec^{2}(x) - \frac{4cos^{2}(x)}{tan(x)} + \frac{4sin^{2}(x)}{tan(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{6cos^{2}(x)sec^{4}(x)}{tan^{3}(x)} - \frac{6sin^{2}(x)sec^{4}(x)}{tan^{3}(x)} - \frac{6sin(x)cos(x)sec^{6}(x)}{tan^{4}(x)} + \frac{10sin(x)cos(x)sec^{4}(x)}{tan^{2}(x)} - \frac{6cos^{2}(x)sec^{2}(x)}{tan(x)} + \frac{6sin^{2}(x)sec^{2}(x)}{tan(x)} + \frac{12sin(x)cos(x)sec^{2}(x)}{tan^{2}(x)} - 4sin(x)cos(x)sec^{2}(x) - \frac{4cos^{2}(x)}{tan(x)} + \frac{4sin^{2}(x)}{tan(x)}\right)}{dx}\\=&\frac{6*-2cos(x)sin(x)sec^{4}(x)}{tan^{3}(x)} + \frac{6cos^{2}(x)*-3sec^{2}(x)(1)sec^{4}(x)}{tan^{4}(x)} + \frac{6cos^{2}(x)*4sec^{4}(x)tan(x)}{tan^{3}(x)} - \frac{6*2sin(x)cos(x)sec^{4}(x)}{tan^{3}(x)} - \frac{6sin^{2}(x)*-3sec^{2}(x)(1)sec^{4}(x)}{tan^{4}(x)} - \frac{6sin^{2}(x)*4sec^{4}(x)tan(x)}{tan^{3}(x)} - \frac{6cos(x)cos(x)sec^{6}(x)}{tan^{4}(x)} - \frac{6sin(x)*-sin(x)sec^{6}(x)}{tan^{4}(x)} - \frac{6sin(x)cos(x)*-4sec^{2}(x)(1)sec^{6}(x)}{tan^{5}(x)} - \frac{6sin(x)cos(x)*6sec^{6}(x)tan(x)}{tan^{4}(x)} + \frac{10cos(x)cos(x)sec^{4}(x)}{tan^{2}(x)} + \frac{10sin(x)*-sin(x)sec^{4}(x)}{tan^{2}(x)} + \frac{10sin(x)cos(x)*-2sec^{2}(x)(1)sec^{4}(x)}{tan^{3}(x)} + \frac{10sin(x)cos(x)*4sec^{4}(x)tan(x)}{tan^{2}(x)} - \frac{6*-2cos(x)sin(x)sec^{2}(x)}{tan(x)} - \frac{6cos^{2}(x)*-sec^{2}(x)(1)sec^{2}(x)}{tan^{2}(x)} - \frac{6cos^{2}(x)*2sec^{2}(x)tan(x)}{tan(x)} + \frac{6*2sin(x)cos(x)sec^{2}(x)}{tan(x)} + \frac{6sin^{2}(x)*-sec^{2}(x)(1)sec^{2}(x)}{tan^{2}(x)} + \frac{6sin^{2}(x)*2sec^{2}(x)tan(x)}{tan(x)} + \frac{12cos(x)cos(x)sec^{2}(x)}{tan^{2}(x)} + \frac{12sin(x)*-sin(x)sec^{2}(x)}{tan^{2}(x)} + \frac{12sin(x)cos(x)*-2sec^{2}(x)(1)sec^{2}(x)}{tan^{3}(x)} + \frac{12sin(x)cos(x)*2sec^{2}(x)tan(x)}{tan^{2}(x)} - 4cos(x)cos(x)sec^{2}(x) - 4sin(x)*-sin(x)sec^{2}(x) - 4sin(x)cos(x)*2sec^{2}(x)tan(x) - \frac{4*-2cos(x)sin(x)}{tan(x)} - \frac{4cos^{2}(x)*-sec^{2}(x)(1)}{tan^{2}(x)} + \frac{4*2sin(x)cos(x)}{tan(x)} + \frac{4sin^{2}(x)*-sec^{2}(x)(1)}{tan^{2}(x)}\\=&\frac{24sin(x)cos(x)sec^{8}(x)}{tan^{5}(x)} - \frac{24cos^{2}(x)sec^{6}(x)}{tan^{4}(x)} + \frac{40cos^{2}(x)sec^{4}(x)}{tan^{2}(x)} - \frac{56sin(x)cos(x)sec^{6}(x)}{tan^{3}(x)} + \frac{24sin^{2}(x)sec^{6}(x)}{tan^{4}(x)} - \frac{40sin^{2}(x)sec^{4}(x)}{tan^{2}(x)} - \frac{48sin(x)cos(x)sec^{4}(x)}{tan^{3}(x)} + \frac{40sin(x)cos(x)sec^{4}(x)}{tan(x)} + \frac{48sin(x)cos(x)sec^{2}(x)}{tan(x)} - 16cos^{2}(x)sec^{2}(x) + 16sin^{2}(x)sec^{2}(x) + \frac{16cos^{2}(x)sec^{2}(x)}{tan^{2}(x)} - \frac{16sin^{2}(x)sec^{2}(x)}{tan^{2}(x)} - 8sin(x)cos(x)tan(x)sec^{2}(x) + \frac{16sin(x)cos(x)}{tan(x)}\\ \end{split}\end{equation} \]

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