There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {lg(x)}^{(\frac{2}{5})}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {lg(x)}^{\frac{2}{5}}\right)}{dx}\\=&({lg(x)}^{\frac{2}{5}}((0)ln(lg(x)) + \frac{(\frac{2}{5})(\frac{1}{ln{10}(x)})}{(lg(x))}))\\=&\frac{2}{5xln{10}lg^{\frac{3}{5}}(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{2}{5xln{10}lg^{\frac{3}{5}}(x)}\right)}{dx}\\=&\frac{2*-1}{5x^{2}ln{10}lg^{\frac{3}{5}}(x)} + \frac{2*-0}{5xln^{2}{10}lg^{\frac{3}{5}}(x)} + \frac{2*\frac{-3}{5}}{5xln{10}lg^{\frac{8}{5}}(x)ln{10}(x)}\\=&\frac{-2}{5x^{2}ln{10}lg^{\frac{3}{5}}(x)} - \frac{6}{25x^{2}ln^{2}{10}lg^{\frac{8}{5}}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2}{5x^{2}ln{10}lg^{\frac{3}{5}}(x)} - \frac{6}{25x^{2}ln^{2}{10}lg^{\frac{8}{5}}(x)}\right)}{dx}\\=&\frac{-2*-2}{5x^{3}ln{10}lg^{\frac{3}{5}}(x)} - \frac{2*-0}{5x^{2}ln^{2}{10}lg^{\frac{3}{5}}(x)} - \frac{2*\frac{-3}{5}}{5x^{2}ln{10}lg^{\frac{8}{5}}(x)ln{10}(x)} - \frac{6*-2}{25x^{3}ln^{2}{10}lg^{\frac{8}{5}}(x)} - \frac{6*-2*0}{25x^{2}ln^{3}{10}lg^{\frac{8}{5}}(x)} - \frac{6*\frac{-8}{5}}{25x^{2}ln^{2}{10}lg^{\frac{13}{5}}(x)ln{10}(x)}\\=&\frac{4}{5x^{3}ln{10}lg^{\frac{3}{5}}(x)} + \frac{18}{25x^{3}ln^{2}{10}lg^{\frac{8}{5}}(x)} + \frac{48}{125x^{3}ln^{3}{10}lg^{\frac{13}{5}}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{4}{5x^{3}ln{10}lg^{\frac{3}{5}}(x)} + \frac{18}{25x^{3}ln^{2}{10}lg^{\frac{8}{5}}(x)} + \frac{48}{125x^{3}ln^{3}{10}lg^{\frac{13}{5}}(x)}\right)}{dx}\\=&\frac{4*-3}{5x^{4}ln{10}lg^{\frac{3}{5}}(x)} + \frac{4*-0}{5x^{3}ln^{2}{10}lg^{\frac{3}{5}}(x)} + \frac{4*\frac{-3}{5}}{5x^{3}ln{10}lg^{\frac{8}{5}}(x)ln{10}(x)} + \frac{18*-3}{25x^{4}ln^{2}{10}lg^{\frac{8}{5}}(x)} + \frac{18*-2*0}{25x^{3}ln^{3}{10}lg^{\frac{8}{5}}(x)} + \frac{18*\frac{-8}{5}}{25x^{3}ln^{2}{10}lg^{\frac{13}{5}}(x)ln{10}(x)} + \frac{48*-3}{125x^{4}ln^{3}{10}lg^{\frac{13}{5}}(x)} + \frac{48*-3*0}{125x^{3}ln^{4}{10}lg^{\frac{13}{5}}(x)} + \frac{48*\frac{-13}{5}}{125x^{3}ln^{3}{10}lg^{\frac{18}{5}}(x)ln{10}(x)}\\=&\frac{-12}{5x^{4}ln{10}lg^{\frac{3}{5}}(x)} - \frac{66}{25x^{4}ln^{2}{10}lg^{\frac{8}{5}}(x)} - \frac{288}{125x^{4}ln^{3}{10}lg^{\frac{13}{5}}(x)} - \frac{624}{625x^{4}ln^{4}{10}lg^{\frac{18}{5}}(x)}\\ \end{split}\end{equation} \]

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