There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{x}{2} + \frac{lg({tan(x)}^{2} + 1)}{4} - log_{5}^{tan(x) + 1}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{1}{2}x + \frac{1}{4}lg(tan^{2}(x) + 1) - log_{5}^{tan(x) + 1}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{1}{2}x + \frac{1}{4}lg(tan^{2}(x) + 1) - log_{5}^{tan(x) + 1}\right)}{dx}\\=&\frac{1}{2} + \frac{\frac{1}{4}(2tan(x)sec^{2}(x)(1) + 0)}{ln{10}(tan^{2}(x) + 1)} - (\frac{(\frac{(sec^{2}(x)(1) + 0)}{(tan(x) + 1)} - \frac{(0)log_{5}^{tan(x) + 1}}{(5)})}{(ln(5))})\\=&\frac{tan(x)sec^{2}(x)}{2(tan^{2}(x) + 1)ln{10}} - \frac{sec^{2}(x)}{(tan(x) + 1)ln(5)} + \frac{1}{2}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{tan(x)sec^{2}(x)}{2(tan^{2}(x) + 1)ln{10}} - \frac{sec^{2}(x)}{(tan(x) + 1)ln(5)} + \frac{1}{2}\right)}{dx}\\=&\frac{(\frac{-(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{2}})tan(x)sec^{2}(x)}{2ln{10}} + \frac{-0tan(x)sec^{2}(x)}{2(tan^{2}(x) + 1)ln^{2}{10}} + \frac{sec^{2}(x)(1)sec^{2}(x)}{2(tan^{2}(x) + 1)ln{10}} + \frac{tan(x)*2sec^{2}(x)tan(x)}{2(tan^{2}(x) + 1)ln{10}} - \frac{(\frac{-(sec^{2}(x)(1) + 0)}{(tan(x) + 1)^{2}})sec^{2}(x)}{ln(5)} - \frac{-0sec^{2}(x)}{(tan(x) + 1)ln^{2}(5)(5)} - \frac{2sec^{2}(x)tan(x)}{(tan(x) + 1)ln(5)} + 0\\=&\frac{-tan^{2}(x)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}ln{10}} + \frac{sec^{4}(x)}{2(tan^{2}(x) + 1)ln{10}} + \frac{tan^{2}(x)sec^{2}(x)}{(tan^{2}(x) + 1)ln{10}} + \frac{sec^{4}(x)}{(tan(x) + 1)^{2}ln(5)} - \frac{2tan(x)sec^{2}(x)}{(tan(x) + 1)ln(5)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-tan^{2}(x)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}ln{10}} + \frac{sec^{4}(x)}{2(tan^{2}(x) + 1)ln{10}} + \frac{tan^{2}(x)sec^{2}(x)}{(tan^{2}(x) + 1)ln{10}} + \frac{sec^{4}(x)}{(tan(x) + 1)^{2}ln(5)} - \frac{2tan(x)sec^{2}(x)}{(tan(x) + 1)ln(5)}\right)}{dx}\\=&\frac{-(\frac{-2(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{3}})tan^{2}(x)sec^{4}(x)}{ln{10}} - \frac{-0tan^{2}(x)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}ln^{2}{10}} - \frac{2tan(x)sec^{2}(x)(1)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}ln{10}} - \frac{tan^{2}(x)*4sec^{4}(x)tan(x)}{(tan^{2}(x) + 1)^{2}ln{10}} + \frac{(\frac{-(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{2}})sec^{4}(x)}{2ln{10}} + \frac{-0sec^{4}(x)}{2(tan^{2}(x) + 1)ln^{2}{10}} + \frac{4sec^{4}(x)tan(x)}{2(tan^{2}(x) + 1)ln{10}} + \frac{(\frac{-(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{2}})tan^{2}(x)sec^{2}(x)}{ln{10}} + \frac{-0tan^{2}(x)sec^{2}(x)}{(tan^{2}(x) + 1)ln^{2}{10}} + \frac{2tan(x)sec^{2}(x)(1)sec^{2}(x)}{(tan^{2}(x) + 1)ln{10}} + \frac{tan^{2}(x)*2sec^{2}(x)tan(x)}{(tan^{2}(x) + 1)ln{10}} + \frac{(\frac{-2(sec^{2}(x)(1) + 0)}{(tan(x) + 1)^{3}})sec^{4}(x)}{ln(5)} + \frac{-0sec^{4}(x)}{(tan(x) + 1)^{2}ln^{2}(5)(5)} + \frac{4sec^{4}(x)tan(x)}{(tan(x) + 1)^{2}ln(5)} - \frac{2(\frac{-(sec^{2}(x)(1) + 0)}{(tan(x) + 1)^{2}})tan(x)sec^{2}(x)}{ln(5)} - \frac{2*-0tan(x)sec^{2}(x)}{(tan(x) + 1)ln^{2}(5)(5)} - \frac{2sec^{2}(x)(1)sec^{2}(x)}{(tan(x) + 1)ln(5)} - \frac{2tan(x)*2sec^{2}(x)tan(x)}{(tan(x) + 1)ln(5)}\\=&\frac{4tan^{3}(x)sec^{6}(x)}{(tan^{2}(x) + 1)^{3}ln{10}} - \frac{3tan(x)sec^{6}(x)}{(tan^{2}(x) + 1)^{2}ln{10}} - \frac{6tan^{3}(x)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}ln{10}} + \frac{4tan(x)sec^{4}(x)}{(tan^{2}(x) + 1)ln{10}} + \frac{2tan^{3}(x)sec^{2}(x)}{(tan^{2}(x) + 1)ln{10}} - \frac{2sec^{6}(x)}{(tan(x) + 1)^{3}ln(5)} + \frac{6tan(x)sec^{4}(x)}{(tan(x) + 1)^{2}ln(5)} - \frac{2sec^{4}(x)}{(tan(x) + 1)ln(5)} - \frac{4tan^{2}(x)sec^{2}(x)}{(tan(x) + 1)ln(5)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{4tan^{3}(x)sec^{6}(x)}{(tan^{2}(x) + 1)^{3}ln{10}} - \frac{3tan(x)sec^{6}(x)}{(tan^{2}(x) + 1)^{2}ln{10}} - \frac{6tan^{3}(x)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}ln{10}} + \frac{4tan(x)sec^{4}(x)}{(tan^{2}(x) + 1)ln{10}} + \frac{2tan^{3}(x)sec^{2}(x)}{(tan^{2}(x) + 1)ln{10}} - \frac{2sec^{6}(x)}{(tan(x) + 1)^{3}ln(5)} + \frac{6tan(x)sec^{4}(x)}{(tan(x) + 1)^{2}ln(5)} - \frac{2sec^{4}(x)}{(tan(x) + 1)ln(5)} - \frac{4tan^{2}(x)sec^{2}(x)}{(tan(x) + 1)ln(5)}\right)}{dx}\\=&\frac{4(\frac{-3(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{4}})tan^{3}(x)sec^{6}(x)}{ln{10}} + \frac{4*-0tan^{3}(x)sec^{6}(x)}{(tan^{2}(x) + 1)^{3}ln^{2}{10}} + \frac{4*3tan^{2}(x)sec^{2}(x)(1)sec^{6}(x)}{(tan^{2}(x) + 1)^{3}ln{10}} + \frac{4tan^{3}(x)*6sec^{6}(x)tan(x)}{(tan^{2}(x) + 1)^{3}ln{10}} - \frac{3(\frac{-2(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{3}})tan(x)sec^{6}(x)}{ln{10}} - \frac{3*-0tan(x)sec^{6}(x)}{(tan^{2}(x) + 1)^{2}ln^{2}{10}} - \frac{3sec^{2}(x)(1)sec^{6}(x)}{(tan^{2}(x) + 1)^{2}ln{10}} - \frac{3tan(x)*6sec^{6}(x)tan(x)}{(tan^{2}(x) + 1)^{2}ln{10}} - \frac{6(\frac{-2(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{3}})tan^{3}(x)sec^{4}(x)}{ln{10}} - \frac{6*-0tan^{3}(x)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}ln^{2}{10}} - \frac{6*3tan^{2}(x)sec^{2}(x)(1)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}ln{10}} - \frac{6tan^{3}(x)*4sec^{4}(x)tan(x)}{(tan^{2}(x) + 1)^{2}ln{10}} + \frac{4(\frac{-(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{2}})tan(x)sec^{4}(x)}{ln{10}} + \frac{4*-0tan(x)sec^{4}(x)}{(tan^{2}(x) + 1)ln^{2}{10}} + \frac{4sec^{2}(x)(1)sec^{4}(x)}{(tan^{2}(x) + 1)ln{10}} + \frac{4tan(x)*4sec^{4}(x)tan(x)}{(tan^{2}(x) + 1)ln{10}} + \frac{2(\frac{-(2tan(x)sec^{2}(x)(1) + 0)}{(tan^{2}(x) + 1)^{2}})tan^{3}(x)sec^{2}(x)}{ln{10}} + \frac{2*-0tan^{3}(x)sec^{2}(x)}{(tan^{2}(x) + 1)ln^{2}{10}} + \frac{2*3tan^{2}(x)sec^{2}(x)(1)sec^{2}(x)}{(tan^{2}(x) + 1)ln{10}} + \frac{2tan^{3}(x)*2sec^{2}(x)tan(x)}{(tan^{2}(x) + 1)ln{10}} - \frac{2(\frac{-3(sec^{2}(x)(1) + 0)}{(tan(x) + 1)^{4}})sec^{6}(x)}{ln(5)} - \frac{2*-0sec^{6}(x)}{(tan(x) + 1)^{3}ln^{2}(5)(5)} - \frac{2*6sec^{6}(x)tan(x)}{(tan(x) + 1)^{3}ln(5)} + \frac{6(\frac{-2(sec^{2}(x)(1) + 0)}{(tan(x) + 1)^{3}})tan(x)sec^{4}(x)}{ln(5)} + \frac{6*-0tan(x)sec^{4}(x)}{(tan(x) + 1)^{2}ln^{2}(5)(5)} + \frac{6sec^{2}(x)(1)sec^{4}(x)}{(tan(x) + 1)^{2}ln(5)} + \frac{6tan(x)*4sec^{4}(x)tan(x)}{(tan(x) + 1)^{2}ln(5)} - \frac{2(\frac{-(sec^{2}(x)(1) + 0)}{(tan(x) + 1)^{2}})sec^{4}(x)}{ln(5)} - \frac{2*-0sec^{4}(x)}{(tan(x) + 1)ln^{2}(5)(5)} - \frac{2*4sec^{4}(x)tan(x)}{(tan(x) + 1)ln(5)} - \frac{4(\frac{-(sec^{2}(x)(1) + 0)}{(tan(x) + 1)^{2}})tan^{2}(x)sec^{2}(x)}{ln(5)} - \frac{4*-0tan^{2}(x)sec^{2}(x)}{(tan(x) + 1)ln^{2}(5)(5)} - \frac{4*2tan(x)sec^{2}(x)(1)sec^{2}(x)}{(tan(x) + 1)ln(5)} - \frac{4tan^{2}(x)*2sec^{2}(x)tan(x)}{(tan(x) + 1)ln(5)}\\=&\frac{-24tan^{4}(x)sec^{8}(x)}{(tan^{2}(x) + 1)^{4}ln{10}} + \frac{24tan^{2}(x)sec^{8}(x)}{(tan^{2}(x) + 1)^{3}ln{10}} + \frac{48tan^{4}(x)sec^{6}(x)}{(tan^{2}(x) + 1)^{3}ln{10}} - \frac{44tan^{2}(x)sec^{6}(x)}{(tan^{2}(x) + 1)^{2}ln{10}} - \frac{3sec^{8}(x)}{(tan^{2}(x) + 1)^{2}ln{10}} + \frac{22tan^{2}(x)sec^{4}(x)}{(tan^{2}(x) + 1)ln{10}} - \frac{16tan(x)sec^{4}(x)}{(tan(x) + 1)ln(5)} - \frac{28tan^{4}(x)sec^{4}(x)}{(tan^{2}(x) + 1)^{2}ln{10}} + \frac{4sec^{6}(x)}{(tan^{2}(x) + 1)ln{10}} + \frac{4tan^{4}(x)sec^{2}(x)}{(tan^{2}(x) + 1)ln{10}} + \frac{6sec^{8}(x)}{(tan(x) + 1)^{4}ln(5)} - \frac{24tan(x)sec^{6}(x)}{(tan(x) + 1)^{3}ln(5)} + \frac{8sec^{6}(x)}{(tan(x) + 1)^{2}ln(5)} + \frac{28tan^{2}(x)sec^{4}(x)}{(tan(x) + 1)^{2}ln(5)} - \frac{8tan^{3}(x)sec^{2}(x)}{(tan(x) + 1)ln(5)}\\ \end{split}\end{equation} \]

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