There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {lg(x)}^{sin(20)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {lg(x)}^{sin(20)}\right)}{dx}\\=&({lg(x)}^{sin(20)}((cos(20)*0)ln(lg(x)) + \frac{(sin(20))(\frac{1}{ln{10}(x)})}{(lg(x))}))\\=&\frac{{lg(x)}^{sin(20)}sin(20)}{xln{10}lg(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{{lg(x)}^{sin(20)}sin(20)}{xln{10}lg(x)}\right)}{dx}\\=&\frac{-{lg(x)}^{sin(20)}sin(20)}{x^{2}ln{10}lg(x)} + \frac{({lg(x)}^{sin(20)}((cos(20)*0)ln(lg(x)) + \frac{(sin(20))(\frac{1}{ln{10}(x)})}{(lg(x))}))sin(20)}{xln{10}lg(x)} + \frac{{lg(x)}^{sin(20)}*-0sin(20)}{xln^{2}{10}lg(x)} + \frac{{lg(x)}^{sin(20)}*-sin(20)}{xln{10}lg^{2}(x)ln{10}(x)} + \frac{{lg(x)}^{sin(20)}cos(20)*0}{xln{10}lg(x)}\\=&\frac{-{lg(x)}^{sin(20)}sin(20)}{x^{2}ln{10}lg(x)} + \frac{{lg(x)}^{sin(20)}sin^{2}(20)}{x^{2}ln^{2}{10}lg^{2}(x)} - \frac{{lg(x)}^{sin(20)}sin(20)}{x^{2}ln^{2}{10}lg^{2}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-{lg(x)}^{sin(20)}sin(20)}{x^{2}ln{10}lg(x)} + \frac{{lg(x)}^{sin(20)}sin^{2}(20)}{x^{2}ln^{2}{10}lg^{2}(x)} - \frac{{lg(x)}^{sin(20)}sin(20)}{x^{2}ln^{2}{10}lg^{2}(x)}\right)}{dx}\\=&\frac{--2{lg(x)}^{sin(20)}sin(20)}{x^{3}ln{10}lg(x)} - \frac{({lg(x)}^{sin(20)}((cos(20)*0)ln(lg(x)) + \frac{(sin(20))(\frac{1}{ln{10}(x)})}{(lg(x))}))sin(20)}{x^{2}ln{10}lg(x)} - \frac{{lg(x)}^{sin(20)}*-0sin(20)}{x^{2}ln^{2}{10}lg(x)} - \frac{{lg(x)}^{sin(20)}*-sin(20)}{x^{2}ln{10}lg^{2}(x)ln{10}(x)} - \frac{{lg(x)}^{sin(20)}cos(20)*0}{x^{2}ln{10}lg(x)} + \frac{-2{lg(x)}^{sin(20)}sin^{2}(20)}{x^{3}ln^{2}{10}lg^{2}(x)} + \frac{({lg(x)}^{sin(20)}((cos(20)*0)ln(lg(x)) + \frac{(sin(20))(\frac{1}{ln{10}(x)})}{(lg(x))}))sin^{2}(20)}{x^{2}ln^{2}{10}lg^{2}(x)} + \frac{{lg(x)}^{sin(20)}*-2*0sin^{2}(20)}{x^{2}ln^{3}{10}lg^{2}(x)} + \frac{{lg(x)}^{sin(20)}*-2sin^{2}(20)}{x^{2}ln^{2}{10}lg^{3}(x)ln{10}(x)} + \frac{{lg(x)}^{sin(20)}*2sin(20)cos(20)*0}{x^{2}ln^{2}{10}lg^{2}(x)} - \frac{-2{lg(x)}^{sin(20)}sin(20)}{x^{3}ln^{2}{10}lg^{2}(x)} - \frac{({lg(x)}^{sin(20)}((cos(20)*0)ln(lg(x)) + \frac{(sin(20))(\frac{1}{ln{10}(x)})}{(lg(x))}))sin(20)}{x^{2}ln^{2}{10}lg^{2}(x)} - \frac{{lg(x)}^{sin(20)}*-2*0sin(20)}{x^{2}ln^{3}{10}lg^{2}(x)} - \frac{{lg(x)}^{sin(20)}*-2sin(20)}{x^{2}ln^{2}{10}lg^{3}(x)ln{10}(x)} - \frac{{lg(x)}^{sin(20)}cos(20)*0}{x^{2}ln^{2}{10}lg^{2}(x)}\\=& - \frac{3{lg(x)}^{sin(20)}sin^{2}(20)}{x^{3}ln^{2}{10}lg^{2}(x)} + \frac{3{lg(x)}^{sin(20)}sin(20)}{x^{3}ln^{2}{10}lg^{2}(x)} + \frac{{lg(x)}^{sin(20)}sin^{3}(20)}{x^{3}ln^{3}{10}lg^{3}(x)} + \frac{2{lg(x)}^{sin(20)}sin(20)}{x^{3}ln{10}lg(x)} - \frac{3{lg(x)}^{sin(20)}sin^{2}(20)}{x^{3}ln^{3}{10}lg^{3}(x)} + \frac{2{lg(x)}^{sin(20)}sin(20)}{x^{3}ln^{3}{10}lg^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - \frac{3{lg(x)}^{sin(20)}sin^{2}(20)}{x^{3}ln^{2}{10}lg^{2}(x)} + \frac{3{lg(x)}^{sin(20)}sin(20)}{x^{3}ln^{2}{10}lg^{2}(x)} + \frac{{lg(x)}^{sin(20)}sin^{3}(20)}{x^{3}ln^{3}{10}lg^{3}(x)} + \frac{2{lg(x)}^{sin(20)}sin(20)}{x^{3}ln{10}lg(x)} - \frac{3{lg(x)}^{sin(20)}sin^{2}(20)}{x^{3}ln^{3}{10}lg^{3}(x)} + \frac{2{lg(x)}^{sin(20)}sin(20)}{x^{3}ln^{3}{10}lg^{3}(x)}\right)}{dx}\\=& - \frac{3*-3{lg(x)}^{sin(20)}sin^{2}(20)}{x^{4}ln^{2}{10}lg^{2}(x)} - \frac{3({lg(x)}^{sin(20)}((cos(20)*0)ln(lg(x)) + \frac{(sin(20))(\frac{1}{ln{10}(x)})}{(lg(x))}))sin^{2}(20)}{x^{3}ln^{2}{10}lg^{2}(x)} - \frac{3{lg(x)}^{sin(20)}*-2*0sin^{2}(20)}{x^{3}ln^{3}{10}lg^{2}(x)} - \frac{3{lg(x)}^{sin(20)}*-2sin^{2}(20)}{x^{3}ln^{2}{10}lg^{3}(x)ln{10}(x)} - \frac{3{lg(x)}^{sin(20)}*2sin(20)cos(20)*0}{x^{3}ln^{2}{10}lg^{2}(x)} + \frac{3*-3{lg(x)}^{sin(20)}sin(20)}{x^{4}ln^{2}{10}lg^{2}(x)} + \frac{3({lg(x)}^{sin(20)}((cos(20)*0)ln(lg(x)) + \frac{(sin(20))(\frac{1}{ln{10}(x)})}{(lg(x))}))sin(20)}{x^{3}ln^{2}{10}lg^{2}(x)} + \frac{3{lg(x)}^{sin(20)}*-2*0sin(20)}{x^{3}ln^{3}{10}lg^{2}(x)} + \frac{3{lg(x)}^{sin(20)}*-2sin(20)}{x^{3}ln^{2}{10}lg^{3}(x)ln{10}(x)} + \frac{3{lg(x)}^{sin(20)}cos(20)*0}{x^{3}ln^{2}{10}lg^{2}(x)} + \frac{-3{lg(x)}^{sin(20)}sin^{3}(20)}{x^{4}ln^{3}{10}lg^{3}(x)} + \frac{({lg(x)}^{sin(20)}((cos(20)*0)ln(lg(x)) + \frac{(sin(20))(\frac{1}{ln{10}(x)})}{(lg(x))}))sin^{3}(20)}{x^{3}ln^{3}{10}lg^{3}(x)} + \frac{{lg(x)}^{sin(20)}*-3*0sin^{3}(20)}{x^{3}ln^{4}{10}lg^{3}(x)} + \frac{{lg(x)}^{sin(20)}*-3sin^{3}(20)}{x^{3}ln^{3}{10}lg^{4}(x)ln{10}(x)} + \frac{{lg(x)}^{sin(20)}*3sin^{2}(20)cos(20)*0}{x^{3}ln^{3}{10}lg^{3}(x)} + \frac{2*-3{lg(x)}^{sin(20)}sin(20)}{x^{4}ln{10}lg(x)} + \frac{2({lg(x)}^{sin(20)}((cos(20)*0)ln(lg(x)) + \frac{(sin(20))(\frac{1}{ln{10}(x)})}{(lg(x))}))sin(20)}{x^{3}ln{10}lg(x)} + \frac{2{lg(x)}^{sin(20)}*-0sin(20)}{x^{3}ln^{2}{10}lg(x)} + \frac{2{lg(x)}^{sin(20)}*-sin(20)}{x^{3}ln{10}lg^{2}(x)ln{10}(x)} + \frac{2{lg(x)}^{sin(20)}cos(20)*0}{x^{3}ln{10}lg(x)} - \frac{3*-3{lg(x)}^{sin(20)}sin^{2}(20)}{x^{4}ln^{3}{10}lg^{3}(x)} - \frac{3({lg(x)}^{sin(20)}((cos(20)*0)ln(lg(x)) + \frac{(sin(20))(\frac{1}{ln{10}(x)})}{(lg(x))}))sin^{2}(20)}{x^{3}ln^{3}{10}lg^{3}(x)} - \frac{3{lg(x)}^{sin(20)}*-3*0sin^{2}(20)}{x^{3}ln^{4}{10}lg^{3}(x)} - \frac{3{lg(x)}^{sin(20)}*-3sin^{2}(20)}{x^{3}ln^{3}{10}lg^{4}(x)ln{10}(x)} - \frac{3{lg(x)}^{sin(20)}*2sin(20)cos(20)*0}{x^{3}ln^{3}{10}lg^{3}(x)} + \frac{2*-3{lg(x)}^{sin(20)}sin(20)}{x^{4}ln^{3}{10}lg^{3}(x)} + \frac{2({lg(x)}^{sin(20)}((cos(20)*0)ln(lg(x)) + \frac{(sin(20))(\frac{1}{ln{10}(x)})}{(lg(x))}))sin(20)}{x^{3}ln^{3}{10}lg^{3}(x)} + \frac{2{lg(x)}^{sin(20)}*-3*0sin(20)}{x^{3}ln^{4}{10}lg^{3}(x)} + \frac{2{lg(x)}^{sin(20)}*-3sin(20)}{x^{3}ln^{3}{10}lg^{4}(x)ln{10}(x)} + \frac{2{lg(x)}^{sin(20)}cos(20)*0}{x^{3}ln^{3}{10}lg^{3}(x)}\\=& - \frac{6{lg(x)}^{sin(20)}sin^{3}(20)}{x^{4}ln^{3}{10}lg^{3}(x)} - \frac{11{lg(x)}^{sin(20)}sin(20)}{x^{4}ln^{2}{10}lg^{2}(x)} + \frac{18{lg(x)}^{sin(20)}sin^{2}(20)}{x^{4}ln^{3}{10}lg^{3}(x)} - \frac{12{lg(x)}^{sin(20)}sin(20)}{x^{4}ln^{3}{10}lg^{3}(x)} + \frac{{lg(x)}^{sin(20)}sin^{4}(20)}{x^{4}ln^{4}{10}lg^{4}(x)} + \frac{11{lg(x)}^{sin(20)}sin^{2}(20)}{x^{4}ln^{2}{10}lg^{2}(x)} - \frac{6{lg(x)}^{sin(20)}sin^{3}(20)}{x^{4}ln^{4}{10}lg^{4}(x)} - \frac{6{lg(x)}^{sin(20)}sin(20)}{x^{4}ln{10}lg(x)} + \frac{11{lg(x)}^{sin(20)}sin^{2}(20)}{x^{4}ln^{4}{10}lg^{4}(x)} - \frac{6{lg(x)}^{sin(20)}sin(20)}{x^{4}ln^{4}{10}lg^{4}(x)}\\ \end{split}\end{equation} \]

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