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

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