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

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