There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{(ln(x) + cos(x))}{tan(x)} + 7\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{ln(x)}{tan(x)} + \frac{cos(x)}{tan(x)} + 7\right)}{dx}\\=&\frac{1}{(x)tan(x)} + \frac{ln(x)*-sec^{2}(x)(1)}{tan^{2}(x)} + \frac{-sin(x)}{tan(x)} + \frac{cos(x)*-sec^{2}(x)(1)}{tan^{2}(x)} + 0\\=&\frac{1}{xtan(x)} - \frac{ln(x)sec^{2}(x)}{tan^{2}(x)} - \frac{sin(x)}{tan(x)} - \frac{cos(x)sec^{2}(x)}{tan^{2}(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{xtan(x)} - \frac{ln(x)sec^{2}(x)}{tan^{2}(x)} - \frac{sin(x)}{tan(x)} - \frac{cos(x)sec^{2}(x)}{tan^{2}(x)}\right)}{dx}\\=&\frac{-1}{x^{2}tan(x)} + \frac{-sec^{2}(x)(1)}{xtan^{2}(x)} - \frac{sec^{2}(x)}{(x)tan^{2}(x)} - \frac{ln(x)*-2sec^{2}(x)(1)sec^{2}(x)}{tan^{3}(x)} - \frac{ln(x)*2sec^{2}(x)tan(x)}{tan^{2}(x)} - \frac{cos(x)}{tan(x)} - \frac{sin(x)*-sec^{2}(x)(1)}{tan^{2}(x)} - \frac{-sin(x)sec^{2}(x)}{tan^{2}(x)} - \frac{cos(x)*-2sec^{2}(x)(1)sec^{2}(x)}{tan^{3}(x)} - \frac{cos(x)*2sec^{2}(x)tan(x)}{tan^{2}(x)}\\=&\frac{-2sec^{2}(x)}{xtan^{2}(x)} - \frac{1}{x^{2}tan(x)} + \frac{2ln(x)sec^{4}(x)}{tan^{3}(x)} - \frac{2ln(x)sec^{2}(x)}{tan(x)} + \frac{2cos(x)sec^{4}(x)}{tan^{3}(x)} + \frac{2sin(x)sec^{2}(x)}{tan^{2}(x)} - \frac{2cos(x)sec^{2}(x)}{tan(x)} - \frac{cos(x)}{tan(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2sec^{2}(x)}{xtan^{2}(x)} - \frac{1}{x^{2}tan(x)} + \frac{2ln(x)sec^{4}(x)}{tan^{3}(x)} - \frac{2ln(x)sec^{2}(x)}{tan(x)} + \frac{2cos(x)sec^{4}(x)}{tan^{3}(x)} + \frac{2sin(x)sec^{2}(x)}{tan^{2}(x)} - \frac{2cos(x)sec^{2}(x)}{tan(x)} - \frac{cos(x)}{tan(x)}\right)}{dx}\\=&\frac{-2*-sec^{2}(x)}{x^{2}tan^{2}(x)} - \frac{2*-2sec^{2}(x)(1)sec^{2}(x)}{xtan^{3}(x)} - \frac{2*2sec^{2}(x)tan(x)}{xtan^{2}(x)} - \frac{-2}{x^{3}tan(x)} - \frac{-sec^{2}(x)(1)}{x^{2}tan^{2}(x)} + \frac{2sec^{4}(x)}{(x)tan^{3}(x)} + \frac{2ln(x)*-3sec^{2}(x)(1)sec^{4}(x)}{tan^{4}(x)} + \frac{2ln(x)*4sec^{4}(x)tan(x)}{tan^{3}(x)} - \frac{2sec^{2}(x)}{(x)tan(x)} - \frac{2ln(x)*-sec^{2}(x)(1)sec^{2}(x)}{tan^{2}(x)} - \frac{2ln(x)*2sec^{2}(x)tan(x)}{tan(x)} + \frac{2*-sin(x)sec^{4}(x)}{tan^{3}(x)} + \frac{2cos(x)*-3sec^{2}(x)(1)sec^{4}(x)}{tan^{4}(x)} + \frac{2cos(x)*4sec^{4}(x)tan(x)}{tan^{3}(x)} + \frac{2cos(x)sec^{2}(x)}{tan^{2}(x)} + \frac{2sin(x)*-2sec^{2}(x)(1)sec^{2}(x)}{tan^{3}(x)} + \frac{2sin(x)*2sec^{2}(x)tan(x)}{tan^{2}(x)} - \frac{2*-sin(x)sec^{2}(x)}{tan(x)} - \frac{2cos(x)*-sec^{2}(x)(1)sec^{2}(x)}{tan^{2}(x)} - \frac{2cos(x)*2sec^{2}(x)tan(x)}{tan(x)} - \frac{-sin(x)}{tan(x)} - \frac{cos(x)*-sec^{2}(x)(1)}{tan^{2}(x)}\\=&\frac{3sec^{2}(x)}{x^{2}tan^{2}(x)} + \frac{6sec^{4}(x)}{xtan^{3}(x)} - \frac{6sec^{2}(x)}{xtan(x)} + \frac{2}{x^{3}tan(x)} - \frac{6ln(x)sec^{6}(x)}{tan^{4}(x)} + \frac{10ln(x)sec^{4}(x)}{tan^{2}(x)} - 4ln(x)sec^{2}(x) - \frac{6sin(x)sec^{4}(x)}{tan^{3}(x)} - \frac{6cos(x)sec^{6}(x)}{tan^{4}(x)} + \frac{10cos(x)sec^{4}(x)}{tan^{2}(x)} + \frac{3cos(x)sec^{2}(x)}{tan^{2}(x)} + \frac{6sin(x)sec^{2}(x)}{tan(x)} - 4cos(x)sec^{2}(x) + \frac{sin(x)}{tan(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3sec^{2}(x)}{x^{2}tan^{2}(x)} + \frac{6sec^{4}(x)}{xtan^{3}(x)} - \frac{6sec^{2}(x)}{xtan(x)} + \frac{2}{x^{3}tan(x)} - \frac{6ln(x)sec^{6}(x)}{tan^{4}(x)} + \frac{10ln(x)sec^{4}(x)}{tan^{2}(x)} - 4ln(x)sec^{2}(x) - \frac{6sin(x)sec^{4}(x)}{tan^{3}(x)} - \frac{6cos(x)sec^{6}(x)}{tan^{4}(x)} + \frac{10cos(x)sec^{4}(x)}{tan^{2}(x)} + \frac{3cos(x)sec^{2}(x)}{tan^{2}(x)} + \frac{6sin(x)sec^{2}(x)}{tan(x)} - 4cos(x)sec^{2}(x) + \frac{sin(x)}{tan(x)}\right)}{dx}\\=&\frac{3*-2sec^{2}(x)}{x^{3}tan^{2}(x)} + \frac{3*-2sec^{2}(x)(1)sec^{2}(x)}{x^{2}tan^{3}(x)} + \frac{3*2sec^{2}(x)tan(x)}{x^{2}tan^{2}(x)} + \frac{6*-sec^{4}(x)}{x^{2}tan^{3}(x)} + \frac{6*-3sec^{2}(x)(1)sec^{4}(x)}{xtan^{4}(x)} + \frac{6*4sec^{4}(x)tan(x)}{xtan^{3}(x)} - \frac{6*-sec^{2}(x)}{x^{2}tan(x)} - \frac{6*-sec^{2}(x)(1)sec^{2}(x)}{xtan^{2}(x)} - \frac{6*2sec^{2}(x)tan(x)}{xtan(x)} + \frac{2*-3}{x^{4}tan(x)} + \frac{2*-sec^{2}(x)(1)}{x^{3}tan^{2}(x)} - \frac{6sec^{6}(x)}{(x)tan^{4}(x)} - \frac{6ln(x)*-4sec^{2}(x)(1)sec^{6}(x)}{tan^{5}(x)} - \frac{6ln(x)*6sec^{6}(x)tan(x)}{tan^{4}(x)} + \frac{10sec^{4}(x)}{(x)tan^{2}(x)} + \frac{10ln(x)*-2sec^{2}(x)(1)sec^{4}(x)}{tan^{3}(x)} + \frac{10ln(x)*4sec^{4}(x)tan(x)}{tan^{2}(x)} - \frac{4sec^{2}(x)}{(x)} - 4ln(x)*2sec^{2}(x)tan(x) - \frac{6cos(x)sec^{4}(x)}{tan^{3}(x)} - \frac{6sin(x)*-3sec^{2}(x)(1)sec^{4}(x)}{tan^{4}(x)} - \frac{6sin(x)*4sec^{4}(x)tan(x)}{tan^{3}(x)} - \frac{6*-sin(x)sec^{6}(x)}{tan^{4}(x)} - \frac{6cos(x)*-4sec^{2}(x)(1)sec^{6}(x)}{tan^{5}(x)} - \frac{6cos(x)*6sec^{6}(x)tan(x)}{tan^{4}(x)} + \frac{10*-sin(x)sec^{4}(x)}{tan^{2}(x)} + \frac{10cos(x)*-2sec^{2}(x)(1)sec^{4}(x)}{tan^{3}(x)} + \frac{10cos(x)*4sec^{4}(x)tan(x)}{tan^{2}(x)} + \frac{3*-sin(x)sec^{2}(x)}{tan^{2}(x)} + \frac{3cos(x)*-2sec^{2}(x)(1)sec^{2}(x)}{tan^{3}(x)} + \frac{3cos(x)*2sec^{2}(x)tan(x)}{tan^{2}(x)} + \frac{6cos(x)sec^{2}(x)}{tan(x)} + \frac{6sin(x)*-sec^{2}(x)(1)sec^{2}(x)}{tan^{2}(x)} + \frac{6sin(x)*2sec^{2}(x)tan(x)}{tan(x)} - 4*-sin(x)sec^{2}(x) - 4cos(x)*2sec^{2}(x)tan(x) + \frac{cos(x)}{tan(x)} + \frac{sin(x)*-sec^{2}(x)(1)}{tan^{2}(x)}\\=&\frac{-8sec^{2}(x)}{x^{3}tan^{2}(x)} - \frac{12sec^{4}(x)}{x^{2}tan^{3}(x)} + \frac{12sec^{2}(x)}{x^{2}tan(x)} - \frac{24sec^{6}(x)}{xtan^{4}(x)} + \frac{40sec^{4}(x)}{xtan^{2}(x)} - \frac{16sec^{2}(x)}{x} - \frac{6}{x^{4}tan(x)} + \frac{24ln(x)sec^{8}(x)}{tan^{5}(x)} - \frac{56ln(x)sec^{6}(x)}{tan^{3}(x)} + \frac{40ln(x)sec^{4}(x)}{tan(x)} - 8ln(x)tan(x)sec^{2}(x) - \frac{12cos(x)sec^{4}(x)}{tan^{3}(x)} + \frac{24sin(x)sec^{6}(x)}{tan^{4}(x)} - \frac{40sin(x)sec^{4}(x)}{tan^{2}(x)} + \frac{24cos(x)sec^{8}(x)}{tan^{5}(x)} - \frac{56cos(x)sec^{6}(x)}{tan^{3}(x)} + \frac{40cos(x)sec^{4}(x)}{tan(x)} - \frac{4sin(x)sec^{2}(x)}{tan^{2}(x)} + \frac{12cos(x)sec^{2}(x)}{tan(x)} + 16sin(x)sec^{2}(x) - 8cos(x)tan(x)sec^{2}(x) + \frac{cos(x)}{tan(x)}\\ \end{split}\end{equation} \]

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