There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ th(lg(x)lg(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = th(lg^{2}(x))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( th(lg^{2}(x))\right)}{dx}\\=&\frac{(1 - th^{2}(lg^{2}(x)))*2lg(x)}{ln{10}(x)}\\=& - \frac{2lg(x)th^{2}(lg^{2}(x))}{xln{10}} + \frac{2lg(x)}{xln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( - \frac{2lg(x)th^{2}(lg^{2}(x))}{xln{10}} + \frac{2lg(x)}{xln{10}}\right)}{dx}\\=& - \frac{2*-lg(x)th^{2}(lg^{2}(x))}{x^{2}ln{10}} - \frac{2*-0lg(x)th^{2}(lg^{2}(x))}{xln^{2}{10}} - \frac{2th^{2}(lg^{2}(x))}{xln{10}ln{10}(x)} - \frac{2lg(x)*2th(lg^{2}(x))(1 - th^{2}(lg^{2}(x)))*2lg(x)}{xln{10}ln{10}(x)} + \frac{2*-lg(x)}{x^{2}ln{10}} + \frac{2*-0lg(x)}{xln^{2}{10}} + \frac{2}{xln{10}ln{10}(x)}\\=&\frac{2lg(x)th^{2}(lg^{2}(x))}{x^{2}ln{10}} - \frac{2th^{2}(lg^{2}(x))}{x^{2}ln^{2}{10}} - \frac{8lg^{2}(x)th(lg^{2}(x))}{x^{2}ln^{2}{10}} + \frac{8lg^{2}(x)th^{3}(lg^{2}(x))}{x^{2}ln^{2}{10}} - \frac{2lg(x)}{x^{2}ln{10}} + \frac{2}{x^{2}ln^{2}{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2lg(x)th^{2}(lg^{2}(x))}{x^{2}ln{10}} - \frac{2th^{2}(lg^{2}(x))}{x^{2}ln^{2}{10}} - \frac{8lg^{2}(x)th(lg^{2}(x))}{x^{2}ln^{2}{10}} + \frac{8lg^{2}(x)th^{3}(lg^{2}(x))}{x^{2}ln^{2}{10}} - \frac{2lg(x)}{x^{2}ln{10}} + \frac{2}{x^{2}ln^{2}{10}}\right)}{dx}\\=&\frac{2*-2lg(x)th^{2}(lg^{2}(x))}{x^{3}ln{10}} + \frac{2*-0lg(x)th^{2}(lg^{2}(x))}{x^{2}ln^{2}{10}} + \frac{2th^{2}(lg^{2}(x))}{x^{2}ln{10}ln{10}(x)} + \frac{2lg(x)*2th(lg^{2}(x))(1 - th^{2}(lg^{2}(x)))*2lg(x)}{x^{2}ln{10}ln{10}(x)} - \frac{2*-2th^{2}(lg^{2}(x))}{x^{3}ln^{2}{10}} - \frac{2*-2*0th^{2}(lg^{2}(x))}{x^{2}ln^{3}{10}} - \frac{2*2th(lg^{2}(x))(1 - th^{2}(lg^{2}(x)))*2lg(x)}{x^{2}ln^{2}{10}ln{10}(x)} - \frac{8*-2lg^{2}(x)th(lg^{2}(x))}{x^{3}ln^{2}{10}} - \frac{8*-2*0lg^{2}(x)th(lg^{2}(x))}{x^{2}ln^{3}{10}} - \frac{8*2lg(x)th(lg^{2}(x))}{x^{2}ln^{2}{10}ln{10}(x)} - \frac{8lg^{2}(x)(1 - th^{2}(lg^{2}(x)))*2lg(x)}{x^{2}ln^{2}{10}ln{10}(x)} + \frac{8*-2lg^{2}(x)th^{3}(lg^{2}(x))}{x^{3}ln^{2}{10}} + \frac{8*-2*0lg^{2}(x)th^{3}(lg^{2}(x))}{x^{2}ln^{3}{10}} + \frac{8*2lg(x)th^{3}(lg^{2}(x))}{x^{2}ln^{2}{10}ln{10}(x)} + \frac{8lg^{2}(x)*3th^{2}(lg^{2}(x))(1 - th^{2}(lg^{2}(x)))*2lg(x)}{x^{2}ln^{2}{10}ln{10}(x)} - \frac{2*-2lg(x)}{x^{3}ln{10}} - \frac{2*-0lg(x)}{x^{2}ln^{2}{10}} - \frac{2}{x^{2}ln{10}ln{10}(x)} + \frac{2*-2}{x^{3}ln^{2}{10}} + \frac{2*-2*0}{x^{2}ln^{3}{10}}\\=& - \frac{4lg(x)th^{2}(lg^{2}(x))}{x^{3}ln{10}} + \frac{6th^{2}(lg^{2}(x))}{x^{3}ln^{2}{10}} + \frac{24lg^{2}(x)th(lg^{2}(x))}{x^{3}ln^{2}{10}} - \frac{24lg^{2}(x)th^{3}(lg^{2}(x))}{x^{3}ln^{2}{10}} - \frac{24lg(x)th(lg^{2}(x))}{x^{3}ln^{3}{10}} + \frac{24lg(x)th^{3}(lg^{2}(x))}{x^{3}ln^{3}{10}} + \frac{64lg^{3}(x)th^{2}(lg^{2}(x))}{x^{3}ln^{3}{10}} - \frac{48lg^{3}(x)th^{4}(lg^{2}(x))}{x^{3}ln^{3}{10}} - \frac{16lg^{3}(x)}{x^{3}ln^{3}{10}} + \frac{4lg(x)}{x^{3}ln{10}} - \frac{6}{x^{3}ln^{2}{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - \frac{4lg(x)th^{2}(lg^{2}(x))}{x^{3}ln{10}} + \frac{6th^{2}(lg^{2}(x))}{x^{3}ln^{2}{10}} + \frac{24lg^{2}(x)th(lg^{2}(x))}{x^{3}ln^{2}{10}} - \frac{24lg^{2}(x)th^{3}(lg^{2}(x))}{x^{3}ln^{2}{10}} - \frac{24lg(x)th(lg^{2}(x))}{x^{3}ln^{3}{10}} + \frac{24lg(x)th^{3}(lg^{2}(x))}{x^{3}ln^{3}{10}} + \frac{64lg^{3}(x)th^{2}(lg^{2}(x))}{x^{3}ln^{3}{10}} - \frac{48lg^{3}(x)th^{4}(lg^{2}(x))}{x^{3}ln^{3}{10}} - \frac{16lg^{3}(x)}{x^{3}ln^{3}{10}} + \frac{4lg(x)}{x^{3}ln{10}} - \frac{6}{x^{3}ln^{2}{10}}\right)}{dx}\\=& - \frac{4*-3lg(x)th^{2}(lg^{2}(x))}{x^{4}ln{10}} - \frac{4*-0lg(x)th^{2}(lg^{2}(x))}{x^{3}ln^{2}{10}} - \frac{4th^{2}(lg^{2}(x))}{x^{3}ln{10}ln{10}(x)} - \frac{4lg(x)*2th(lg^{2}(x))(1 - th^{2}(lg^{2}(x)))*2lg(x)}{x^{3}ln{10}ln{10}(x)} + \frac{6*-3th^{2}(lg^{2}(x))}{x^{4}ln^{2}{10}} + \frac{6*-2*0th^{2}(lg^{2}(x))}{x^{3}ln^{3}{10}} + \frac{6*2th(lg^{2}(x))(1 - th^{2}(lg^{2}(x)))*2lg(x)}{x^{3}ln^{2}{10}ln{10}(x)} + \frac{24*-3lg^{2}(x)th(lg^{2}(x))}{x^{4}ln^{2}{10}} + \frac{24*-2*0lg^{2}(x)th(lg^{2}(x))}{x^{3}ln^{3}{10}} + \frac{24*2lg(x)th(lg^{2}(x))}{x^{3}ln^{2}{10}ln{10}(x)} + \frac{24lg^{2}(x)(1 - th^{2}(lg^{2}(x)))*2lg(x)}{x^{3}ln^{2}{10}ln{10}(x)} - \frac{24*-3lg^{2}(x)th^{3}(lg^{2}(x))}{x^{4}ln^{2}{10}} - \frac{24*-2*0lg^{2}(x)th^{3}(lg^{2}(x))}{x^{3}ln^{3}{10}} - \frac{24*2lg(x)th^{3}(lg^{2}(x))}{x^{3}ln^{2}{10}ln{10}(x)} - \frac{24lg^{2}(x)*3th^{2}(lg^{2}(x))(1 - th^{2}(lg^{2}(x)))*2lg(x)}{x^{3}ln^{2}{10}ln{10}(x)} - \frac{24*-3lg(x)th(lg^{2}(x))}{x^{4}ln^{3}{10}} - \frac{24*-3*0lg(x)th(lg^{2}(x))}{x^{3}ln^{4}{10}} - \frac{24th(lg^{2}(x))}{x^{3}ln^{3}{10}ln{10}(x)} - \frac{24lg(x)(1 - th^{2}(lg^{2}(x)))*2lg(x)}{x^{3}ln^{3}{10}ln{10}(x)} + \frac{24*-3lg(x)th^{3}(lg^{2}(x))}{x^{4}ln^{3}{10}} + \frac{24*-3*0lg(x)th^{3}(lg^{2}(x))}{x^{3}ln^{4}{10}} + \frac{24th^{3}(lg^{2}(x))}{x^{3}ln^{3}{10}ln{10}(x)} + \frac{24lg(x)*3th^{2}(lg^{2}(x))(1 - th^{2}(lg^{2}(x)))*2lg(x)}{x^{3}ln^{3}{10}ln{10}(x)} + \frac{64*-3lg^{3}(x)th^{2}(lg^{2}(x))}{x^{4}ln^{3}{10}} + \frac{64*-3*0lg^{3}(x)th^{2}(lg^{2}(x))}{x^{3}ln^{4}{10}} + \frac{64*3lg^{2}(x)th^{2}(lg^{2}(x))}{x^{3}ln^{3}{10}ln{10}(x)} + \frac{64lg^{3}(x)*2th(lg^{2}(x))(1 - th^{2}(lg^{2}(x)))*2lg(x)}{x^{3}ln^{3}{10}ln{10}(x)} - \frac{48*-3lg^{3}(x)th^{4}(lg^{2}(x))}{x^{4}ln^{3}{10}} - \frac{48*-3*0lg^{3}(x)th^{4}(lg^{2}(x))}{x^{3}ln^{4}{10}} - \frac{48*3lg^{2}(x)th^{4}(lg^{2}(x))}{x^{3}ln^{3}{10}ln{10}(x)} - \frac{48lg^{3}(x)*4th^{3}(lg^{2}(x))(1 - th^{2}(lg^{2}(x)))*2lg(x)}{x^{3}ln^{3}{10}ln{10}(x)} - \frac{16*-3lg^{3}(x)}{x^{4}ln^{3}{10}} - \frac{16*-3*0lg^{3}(x)}{x^{3}ln^{4}{10}} - \frac{16*3lg^{2}(x)}{x^{3}ln^{3}{10}ln{10}(x)} + \frac{4*-3lg(x)}{x^{4}ln{10}} + \frac{4*-0lg(x)}{x^{3}ln^{2}{10}} + \frac{4}{x^{3}ln{10}ln{10}(x)} - \frac{6*-3}{x^{4}ln^{2}{10}} - \frac{6*-2*0}{x^{3}ln^{3}{10}}\\=&\frac{12lg(x)th^{2}(lg^{2}(x))}{x^{4}ln{10}} - \frac{22th^{2}(lg^{2}(x))}{x^{4}ln^{2}{10}} - \frac{88lg^{2}(x)th(lg^{2}(x))}{x^{4}ln^{2}{10}} + \frac{88lg^{2}(x)th^{3}(lg^{2}(x))}{x^{4}ln^{2}{10}} + \frac{144lg(x)th(lg^{2}(x))}{x^{4}ln^{3}{10}} - \frac{144lg(x)th^{3}(lg^{2}(x))}{x^{4}ln^{3}{10}} - \frac{384lg^{3}(x)th^{2}(lg^{2}(x))}{x^{4}ln^{3}{10}} + \frac{288lg^{3}(x)th^{4}(lg^{2}(x))}{x^{4}ln^{3}{10}} + \frac{384lg^{2}(x)th^{2}(lg^{2}(x))}{x^{4}ln^{4}{10}} - \frac{288lg^{2}(x)th^{4}(lg^{2}(x))}{x^{4}ln^{4}{10}} - \frac{24th(lg^{2}(x))}{x^{4}ln^{4}{10}} + \frac{256lg^{4}(x)th(lg^{2}(x))}{x^{4}ln^{4}{10}} - \frac{640lg^{4}(x)th^{3}(lg^{2}(x))}{x^{4}ln^{4}{10}} + \frac{24th^{3}(lg^{2}(x))}{x^{4}ln^{4}{10}} + \frac{384lg^{4}(x)th^{5}(lg^{2}(x))}{x^{4}ln^{4}{10}} - \frac{96lg^{2}(x)}{x^{4}ln^{4}{10}} + \frac{96lg^{3}(x)}{x^{4}ln^{3}{10}} - \frac{12lg(x)}{x^{4}ln{10}} + \frac{22}{x^{4}ln^{2}{10}}\\ \end{split}\end{equation} \]

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