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

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