There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ log_{(400 - x)}^{1002 - x}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = log_{-x + 400}^{-x + 1002}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{-x + 400}^{-x + 1002}\right)}{dx}\\=&(\frac{(\frac{(-1 + 0)}{(-x + 1002)} - \frac{(-1 + 0)log_{-x + 400}^{-x + 1002}}{(-x + 400)})}{(ln(-x + 400))})\\=&\frac{log_{-x + 400}^{-x + 1002}}{(-x + 400)ln(-x + 400)} - \frac{1}{(-x + 1002)ln(-x + 400)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{log_{-x + 400}^{-x + 1002}}{(-x + 400)ln(-x + 400)} - \frac{1}{(-x + 1002)ln(-x + 400)}\right)}{dx}\\=&\frac{(\frac{-(-1 + 0)}{(-x + 400)^{2}})log_{-x + 400}^{-x + 1002}}{ln(-x + 400)} + \frac{(\frac{(\frac{(-1 + 0)}{(-x + 1002)} - \frac{(-1 + 0)log_{-x + 400}^{-x + 1002}}{(-x + 400)})}{(ln(-x + 400))})}{(-x + 400)ln(-x + 400)} + \frac{log_{-x + 400}^{-x + 1002}*-(-1 + 0)}{(-x + 400)ln^{2}(-x + 400)(-x + 400)} - \frac{(\frac{-(-1 + 0)}{(-x + 1002)^{2}})}{ln(-x + 400)} - \frac{-(-1 + 0)}{(-x + 1002)ln^{2}(-x + 400)(-x + 400)}\\=&\frac{log_{-x + 400}^{-x + 1002}}{(-x + 400)^{2}ln(-x + 400)} + \frac{2log_{-x + 400}^{-x + 1002}}{(-x + 400)^{2}ln^{2}(-x + 400)} - \frac{2}{(-x + 400)(-x + 1002)ln^{2}(-x + 400)} - \frac{1}{(-x + 1002)^{2}ln(-x + 400)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{log_{-x + 400}^{-x + 1002}}{(-x + 400)^{2}ln(-x + 400)} + \frac{2log_{-x + 400}^{-x + 1002}}{(-x + 400)^{2}ln^{2}(-x + 400)} - \frac{2}{(-x + 400)(-x + 1002)ln^{2}(-x + 400)} - \frac{1}{(-x + 1002)^{2}ln(-x + 400)}\right)}{dx}\\=&\frac{(\frac{-2(-1 + 0)}{(-x + 400)^{3}})log_{-x + 400}^{-x + 1002}}{ln(-x + 400)} + \frac{(\frac{(\frac{(-1 + 0)}{(-x + 1002)} - \frac{(-1 + 0)log_{-x + 400}^{-x + 1002}}{(-x + 400)})}{(ln(-x + 400))})}{(-x + 400)^{2}ln(-x + 400)} + \frac{log_{-x + 400}^{-x + 1002}*-(-1 + 0)}{(-x + 400)^{2}ln^{2}(-x + 400)(-x + 400)} + \frac{2(\frac{-2(-1 + 0)}{(-x + 400)^{3}})log_{-x + 400}^{-x + 1002}}{ln^{2}(-x + 400)} + \frac{2(\frac{(\frac{(-1 + 0)}{(-x + 1002)} - \frac{(-1 + 0)log_{-x + 400}^{-x + 1002}}{(-x + 400)})}{(ln(-x + 400))})}{(-x + 400)^{2}ln^{2}(-x + 400)} + \frac{2log_{-x + 400}^{-x + 1002}*-2(-1 + 0)}{(-x + 400)^{2}ln^{3}(-x + 400)(-x + 400)} - \frac{2(\frac{-(-1 + 0)}{(-x + 400)^{2}})}{(-x + 1002)ln^{2}(-x + 400)} - \frac{2(\frac{-(-1 + 0)}{(-x + 1002)^{2}})}{(-x + 400)ln^{2}(-x + 400)} - \frac{2*-2(-1 + 0)}{(-x + 400)(-x + 1002)ln^{3}(-x + 400)(-x + 400)} - \frac{(\frac{-2(-1 + 0)}{(-x + 1002)^{3}})}{ln(-x + 400)} - \frac{-(-1 + 0)}{(-x + 1002)^{2}ln^{2}(-x + 400)(-x + 400)}\\=&\frac{2log_{-x + 400}^{-x + 1002}}{(-x + 400)^{3}ln(-x + 400)} + \frac{6log_{-x + 400}^{-x + 1002}}{(-x + 400)^{3}ln^{2}(-x + 400)} - \frac{3}{(-x + 400)^{2}(-x + 1002)ln^{2}(-x + 400)} + \frac{6log_{-x + 400}^{-x + 1002}}{(-x + 400)^{3}ln^{3}(-x + 400)} - \frac{6}{(-x + 400)^{2}(-x + 1002)ln^{3}(-x + 400)} - \frac{2}{(-x + 1002)^{2}(-x + 400)ln^{2}(-x + 400)} - \frac{2}{(-x + 1002)^{3}ln(-x + 400)} - \frac{1}{(-x + 400)(-x + 1002)^{2}ln^{2}(-x + 400)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2log_{-x + 400}^{-x + 1002}}{(-x + 400)^{3}ln(-x + 400)} + \frac{6log_{-x + 400}^{-x + 1002}}{(-x + 400)^{3}ln^{2}(-x + 400)} - \frac{3}{(-x + 400)^{2}(-x + 1002)ln^{2}(-x + 400)} + \frac{6log_{-x + 400}^{-x + 1002}}{(-x + 400)^{3}ln^{3}(-x + 400)} - \frac{6}{(-x + 400)^{2}(-x + 1002)ln^{3}(-x + 400)} - \frac{2}{(-x + 1002)^{2}(-x + 400)ln^{2}(-x + 400)} - \frac{2}{(-x + 1002)^{3}ln(-x + 400)} - \frac{1}{(-x + 400)(-x + 1002)^{2}ln^{2}(-x + 400)}\right)}{dx}\\=&\frac{2(\frac{-3(-1 + 0)}{(-x + 400)^{4}})log_{-x + 400}^{-x + 1002}}{ln(-x + 400)} + \frac{2(\frac{(\frac{(-1 + 0)}{(-x + 1002)} - \frac{(-1 + 0)log_{-x + 400}^{-x + 1002}}{(-x + 400)})}{(ln(-x + 400))})}{(-x + 400)^{3}ln(-x + 400)} + \frac{2log_{-x + 400}^{-x + 1002}*-(-1 + 0)}{(-x + 400)^{3}ln^{2}(-x + 400)(-x + 400)} + \frac{6(\frac{-3(-1 + 0)}{(-x + 400)^{4}})log_{-x + 400}^{-x + 1002}}{ln^{2}(-x + 400)} + \frac{6(\frac{(\frac{(-1 + 0)}{(-x + 1002)} - \frac{(-1 + 0)log_{-x + 400}^{-x + 1002}}{(-x + 400)})}{(ln(-x + 400))})}{(-x + 400)^{3}ln^{2}(-x + 400)} + \frac{6log_{-x + 400}^{-x + 1002}*-2(-1 + 0)}{(-x + 400)^{3}ln^{3}(-x + 400)(-x + 400)} - \frac{3(\frac{-2(-1 + 0)}{(-x + 400)^{3}})}{(-x + 1002)ln^{2}(-x + 400)} - \frac{3(\frac{-(-1 + 0)}{(-x + 1002)^{2}})}{(-x + 400)^{2}ln^{2}(-x + 400)} - \frac{3*-2(-1 + 0)}{(-x + 400)^{2}(-x + 1002)ln^{3}(-x + 400)(-x + 400)} + \frac{6(\frac{-3(-1 + 0)}{(-x + 400)^{4}})log_{-x + 400}^{-x + 1002}}{ln^{3}(-x + 400)} + \frac{6(\frac{(\frac{(-1 + 0)}{(-x + 1002)} - \frac{(-1 + 0)log_{-x + 400}^{-x + 1002}}{(-x + 400)})}{(ln(-x + 400))})}{(-x + 400)^{3}ln^{3}(-x + 400)} + \frac{6log_{-x + 400}^{-x + 1002}*-3(-1 + 0)}{(-x + 400)^{3}ln^{4}(-x + 400)(-x + 400)} - \frac{6(\frac{-2(-1 + 0)}{(-x + 400)^{3}})}{(-x + 1002)ln^{3}(-x + 400)} - \frac{6(\frac{-(-1 + 0)}{(-x + 1002)^{2}})}{(-x + 400)^{2}ln^{3}(-x + 400)} - \frac{6*-3(-1 + 0)}{(-x + 400)^{2}(-x + 1002)ln^{4}(-x + 400)(-x + 400)} - \frac{2(\frac{-2(-1 + 0)}{(-x + 1002)^{3}})}{(-x + 400)ln^{2}(-x + 400)} - \frac{2(\frac{-(-1 + 0)}{(-x + 400)^{2}})}{(-x + 1002)^{2}ln^{2}(-x + 400)} - \frac{2*-2(-1 + 0)}{(-x + 1002)^{2}(-x + 400)ln^{3}(-x + 400)(-x + 400)} - \frac{2(\frac{-3(-1 + 0)}{(-x + 1002)^{4}})}{ln(-x + 400)} - \frac{2*-(-1 + 0)}{(-x + 1002)^{3}ln^{2}(-x + 400)(-x + 400)} - \frac{(\frac{-(-1 + 0)}{(-x + 400)^{2}})}{(-x + 1002)^{2}ln^{2}(-x + 400)} - \frac{(\frac{-2(-1 + 0)}{(-x + 1002)^{3}})}{(-x + 400)ln^{2}(-x + 400)} - \frac{-2(-1 + 0)}{(-x + 400)(-x + 1002)^{2}ln^{3}(-x + 400)(-x + 400)}\\=&\frac{6log_{-x + 400}^{-x + 1002}}{(-x + 400)^{4}ln(-x + 400)} + \frac{22log_{-x + 400}^{-x + 1002}}{(-x + 400)^{4}ln^{2}(-x + 400)} - \frac{8}{(-x + 400)^{3}(-x + 1002)ln^{2}(-x + 400)} + \frac{36log_{-x + 400}^{-x + 1002}}{(-x + 400)^{4}ln^{3}(-x + 400)} - \frac{24}{(-x + 400)^{3}(-x + 1002)ln^{3}(-x + 400)} - \frac{3}{(-x + 1002)^{2}(-x + 400)^{2}ln^{2}(-x + 400)} + \frac{24log_{-x + 400}^{-x + 1002}}{(-x + 400)^{4}ln^{4}(-x + 400)} - \frac{24}{(-x + 400)^{3}(-x + 1002)ln^{4}(-x + 400)} - \frac{10}{(-x + 1002)^{2}(-x + 400)^{2}ln^{3}(-x + 400)} - \frac{6}{(-x + 1002)^{3}(-x + 400)ln^{2}(-x + 400)} - \frac{3}{(-x + 400)^{2}(-x + 1002)^{2}ln^{2}(-x + 400)} - \frac{6}{(-x + 1002)^{4}ln(-x + 400)} - \frac{2}{(-x + 400)(-x + 1002)^{3}ln^{2}(-x + 400)} - \frac{2}{(-x + 400)^{2}(-x + 1002)^{2}ln^{3}(-x + 400)}\\ \end{split}\end{equation} \]

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